A History of Mathematical Notations/Volume 1/Babylonians

by Florian Cajori

II

NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS

BABYLONIANS

1. In the Babylonian notation of numbers a vertical wedge 𒁹 stood for 1, while the characters 𒌋 and 𒈨 signified 10 and 100, respectively. Grotefend^[1] believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out. Ordinarily, two principles were employed in the Babylonial notation—the additive and multiplicative. We shall see that limited use was made of a third principle, that of subtraction.

2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, 𒈨𒎙𒐈 stands for 123. The principle of multiplication reveals itself in 𒌋𒈨 where the smaller symbol 10, placed before the 100, is to be multiplied by 100, so that this symbolism designates 1,000.

3. These cuneiform symbols were probably invented by the early Sumerians. Their inscriptions disclose the use of a decimal scale of numbers and also of a sexagesimal scale.^[2]

Early Sumerian clay tablets contain also numerals expressed by circles and curved signs, made with the blunt circular end of a stylus, the ordinary wedge-shaped characters being made with the pointed end. A circle ● stood for 10, a semicircular or lunar sign stood for 1. Thus, a "round-up" of cattle shows ● ●
● ᗞᗞᗞ
ᗞᗞᗞ, or 36, cows.^[3]

4. The sexagesimal scale was first discovered on a tablet by E. Hincks^[4] in 1854. It records the magnitude of the illuminated portion of the moon's disk for every day from new to full moon, the whole disk being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20, which is a geometrical progression, on the assumption that the last number is 80. From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the common difference being 16. The last number is written in the tablet 𒃻, and, according to Hincks's interpretation, stood for 4×60=240.

Fig. 1.—Babylonian tablets from Nippur, about 2400 B.C.

5. Hincks's explanation was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854, and called the Tablets of Senkereh. One tablet was found to contain a table of square numbers, from 1² to 60², a second one a table of cube numbers from 1³ to 32³. The tablets were probably written between 2300 and 1600 B.C. Various scholars contributed toward their interpretation. Among them were George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr. Lenormant, and finally R. Lepsius.^[5] The numbers 1, 4, 9, 16, 25, 36, and 49 are given as the squares of the first seven integers, respectively. We have next 1.4=8², 1.21=9², 1.40=10², etc. This clearly indicates the use of the sexagesimal scale which makes 1.4=60+4, 1.21=60+21, 1.40=60+40, etc. This sexagesimal system marks the earliest appearance of the all-important "principle of position" in writing numbers. In its general and systematic application, this principle requires a symbol for zero. But no such symbol has been found on early Babylonian tablets; records of about 200 B.C. give a symbol for zero as we shall see later, but it was not used in calculation. The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era.

6. An extension of our knowledge of Babylonian mathematics was made by H. V. Hilprecht who made excavations at Nuffar (the ancient Nippur). We reproduce one of his tablets^[6] in Figure 1.

Hilprecht's transliteration, as given on page 28 of his text is as follows:

Line 1.	125	720
Line 2.	IGI-GAL-BI	103,680
Line 3.	250	360
Line 4.	IGI-GAL-BI	51,840
Line 5.	500	180
Line 6.	IGI-GAL-BI	25,920
Line 7.	1,000	90
Line 8.	IGI-GAL-BI	12,960
Line 9.	2,000	18
Line 10.	IGI-GAL-BI	6,480
Line 11.	4,000	9
Line 12.	IGI-GAL-BI	3,240
Line 13.	8,000	18
Line 14.	IGI-GAL-BI	1,620
Line 15.	16,000	9
Line 16.	IGI-GAL-BI	810

7. In further explanation, observe that in

Line 1.	125=2×60+5,	720=12×60+0
Line 2.	Its denominator,	103,680=[28×60+48(?)]×60+0
Line 3.	250=4×60+10,	360=6×60+0
Line 4.	Its denominator,	51,840=[14×60+24]×60+0
Line 5.	500=8×60+20,	180=3×60+0
Line 6.	Its denominator,	25,920=[7×60+12]×60+0
Line 7.	1,000=16×60+40,	90=1×60+30
Line 8.	Its denominator,	12,960=[3×60+36]×60+0
Line 9.	2,000=33×60+20,	18=10+8
Line 10.	Its denominator,	6,480 = [1×60+48]×60+0
Line 11.	4,000=[1×60+6]×60+40,	9
Line 12.	Its denominator,	3,240 = 54 X 60+0
Line 13.	8,000=[2×60+13]×60+20,	18
Line 14.	Its denominator,	1,620=27×60+0
Line 15.	16,000=[4×60+26]×60+40,	9
Line 16.	Its denominator,	810=13×60+30
IGI-GAL = Denominator, BI = Its, i.e., the number 12,960,000 or 604.

We quote from Hilprecht (op. cit., pp. 28–30):

"We observe (a) that the first numbers of all the odd lines (1, 3, 5, 7, 9, 11, 13, 15) form an increasing, and all the numbers of the even lines (preceded by IGI-GAL-BI = 'its denominator') a descending geometrical progression; (b) that the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denominator, in other words,

16,000125=12,960,000/103,680; 16,000250=12,960,000/51,840; 16,000500=12,960,000/25,920; 161,000=12,960,000/12,960; 162,000=12,960,000/6,480; 164,000=12,960,000/3,240; 168,000=12,960,000/1,620; 16,000=12,960,000/810.

But the closing numbers of all the odd lines (720, 360, 180, 90, 18, 9, 18, 9) are still obscure to me.....

"The question arises, what is the meaning of all this? What in particular is the meaning of the number 12,960,000 (=60⁴ or 3,600²) which underlies all the mathematical texts here treated....?.... This 'geometrical number' (12,960,000), which he [Plato in his Republic viii. 546B–D] calls 'the lord of better and worse births,' is the arithmetical expression of a great law controlling the Universe. According to Adam this law is 'the Law of Change, that law of inevitable degeneration to which the Universe and all its parts are subject'—an interpretation from which I am obliged to differ. On the contrary, it is the Law of Uniformity or Harmony, i.e. that fundamental law which governs the Universe and all its parts, and which cannot be ignored and violated without causing an anomaly, i.e. without resulting in a degeneration of the race." The nature of the "Platonic number" is still a debated question.

8. In the reading of numbers expressed in the Babylonian sexagesimal system, uncertainty arises from the fact that the early Babylonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is “sixties” and not “units.” Only from the study of the entire tablet has it been inferred that the number intended is 12×60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence of units of lower denomination. But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places.

9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, 36,000.^[7] Thus · was 10, ● was 3,600, ⦾ was 36,000.

⁠in view of other variants occurring in the mathematical tablets from Nippur, notably the numerous variants of “19,”¹ some of which may be merely scribal errors:

They evidently all go back to the form 𒎙𒇲𒁹 or 𒎙𒇳𒁹 (20−1=19).

Fig. 2.—Showing application of the principle of subtraction

10. Besides the principles of addition and multiplication, Babylonian tablets reveal also the use of the principle of subtraction, which is familiar to us in the Roman notation XIX (20−1) for the number 19. Hilprecht has collected ideograms from the Babylonian tablets which he has studied, which represent the number 19. We reproduce his symbols in Figure 2. In each of these twelve ideograms (Fig. 2), the two symbols to the left signify together 20. Of the symbols immediately to the right of the 20, one vertical wedge stands for “one” and the remaining symbols, for instance 𒇲, for LAL or “minus”; the entire ideogram represents in each of the twelve cases the number 20−1 or 19.

One finds the principle of subtraction used also with curved signs;^[8] ᗞ●●𒇲ᗞ meant 60+20−1, or 79.

11. The symbol used about the second century B.C. to designate absence of a number, or a blank space, is shown in Figure 3, containing numerical data relating to the moon.^[9] As previously stated, this symbol, 𒑊, was not used in computation and therefore performed only a small part of the functions of our modern zero. The symbol is an in the tablet in row 10, column 12; also in row 8, column 13. Kugler's translation of the tablet, given in his book, page 42, is shown below. Of the last column only an indistinct fragment is preserved; the rest is broken off.

Fig. 3.—Babylonian lunar tables, reverse; full moon for one year, about the end of the second century b.c.

REVERSE

1...	Nisannu	28°56′30″	19°16′00″	Librae	3z06°45′	4I74II10III	sik
2...	Airu	28°38′30″	17°54′30″	Scorpii	3z21°28′	6I20II30III	sik
3...	Simannu	28°20′30″	16°15′00″	Arcitenentis	3z31°39′	3I45II30III	sik
4...	Dûzu	28°18′30″	| 14°33′30″	Capri	3z34°41′	1I10II30III	sik
5...	Âbu	28°36′30″	13°09′00″	Aquarii	3z27°56′	1I24II30III	bar
6...	Ulûlu	29°54′30″	13°03′30″	Piscium	3z15°34′	1I59II30III	num
7...	Tišrîtu	29°12′30″	11°16′00″	Arietis	2z58°03′	4I34II30III	num
8...	Araḫ-s.	29°30′30″	10°46′30″	Tauri	2z40°54′	6I00II10III	num
9...	Kislimu	29°48′30″	10°35′00″	Geminorum	2z29°29′	3I25II10III	num
10...	Tebitu	29°57′30″	10°32′30″	Cancri	2z24°30′	0I57II10III	num
11...	Šabâtu	29°39′30″	10°12′00″	Leonis	2z30°53′	1I44II50III	bar
12...	Adâru I	29°21′30″	09°33′30″	Virginis	2z42°56′	2I19II50III	sik
13...	Adâru II	29°03′30″	08°36′00″	Librae	3z00°21′	4I54II50III	sik
14...	Nisannu	28°45′30″	07°21′30″	Scorpii	3z17°36′	5I39II50III	sik

Fig. 4.—Mathematical cuneiform tablet, CBS 8536, in the Museum of the University of Pennsylvania.

12. J. Oppert pointed out the Babylonian use of a designation for the sixths, viz., ⅙, ⅓, ½, ⅔, ⅚. These are unit fractions or fractions whose numerators are one less than the denominators.^[10] He also advanced evidence pointing to the Babylonian use of sexagesimal fractions and the use of the sexagesimal system in weights and measures. The occurrence of sexagesimal fractions is shown in tablets recently examined. We reproduce in Figure 4 two out of twelve columns found on a tablet described by H. F. Lutz.^[11] According to Lutz, the tablet “cannot be placed later than the Cassite period, but it seems more probable that it goes back even to the First Dynasty period, ca. 2000 B.C.”

13. To mathematicians the tablet is of interest because it reveals operations with sexagesimal fractions resembling modern operations with decimal fractions. For example, 60 is divided by 81 and the quotient expressed sexagesimally. Again, a sexagesimal number with two fractional places, 44(26)(40), is multiplied by itself, yielding a product in four fractional places, namely, [32]55(18)(31)(6)(40). In this notation the [32] stands for 32×60 units, and to the (18), (31), (6), (40) must be assigned, respectively, the denominators 60, 60², 60³, 60⁴.

The tablet contains twelve columns of figures. The first column (Fig. 4) gives the results of dividing 60 in succession by twenty-nine different divisors from 2 to 81. The eleven other columns contain tables of multiplication; each of the numbers 50, 48, 45, 44(26)(40), 40, 36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by the numbers 30, 40, 50, and finally by itself. Using our modern numerals, we interpret on page 10 the first and the fifth columns. They exhibit a larger number of fractions than do the other columns. The Babylonians had no mark separating the fractional from the integral parts of a number. Hence a number like 44(26)(40) might be interpreted in different ways; among the possible meanings are 44×60²+26×60+40, 44×6O+26+40×60⁻¹, and 44+26×60⁻¹+40×60⁻². Which interpretation is the correct one can be judged only by the context, if at all.

The exact meaning of the first two lines in the first column is uncertain. In this column 60 is divided by each of the integers written on the left. The respective quotients are placed on the right.

In the fifth column the multiplicand is 44(26)(40) or 44 4/9. The last two lines seem to mean "60²÷44(26)(40)=81, 60²÷81=44(26)(40)."

⁠First Column . . . . gal (?) -bi 40 -ám šu a- na gal-bi 30 -ám			Fifth Column
			44(26)(40)
igi 2	30	1	44(26)(40)
igi 3	20	2	[1]28(53)(20)
igi 4	15	3	[2]13(20)
igi 5	12	4	[2]48(56)(40)*
igi 6	10	5	[3]42(13)(20)
igi 8	7(30)	6	[4]26(40)
igi 9	6(40)	7	[5]11(6)(40)
igi 10	6	9	[6]40
igi 12	5	10	[7]24(26)(40)
igi 15	4	11	[8]8(53)(20)
igi 16	3(45)	12	[8]53(20)
igi 18	3(20)	13	[9]27(46)(40)*
igi 20	3	14	[10]22(13)(20)
igi 24	2(30)	15	[11]6(40)
igi 25	2(24)	16	[11]51(6)(40)
igi 28*⁠	2(13)(20)	17	[12]35(33)(20)
igi 30	2	18	[13]20
igi 35*	1(52)(30)	19	[14]4(26)(40)
igi 36	1(40)	20	[14]48(53)(20)
igi 40	1(30)	30	[[22]13(20)
igi 45	1(20)	40	[29]37(46)(40)
igi 48	1(15)	50	[38]2(13)(20)*
igi 50	1(12)	44(26)(40)a-na 44(26)(40)
igi 54	1(6)(40)	[32]55(18)(31)(6)(40)
igi 60	1	44(26)(40) square
igi 64	(56)(15)	igi 44(26)(40)	81
igi 72	(50)	igi 81	44(26)(40)
igi 80	(45)
igi 81	(44)(26)(40)⁠⁠
Numbers that are incorrect are marked by an asterisk (*).

14. The Babylonian use of sexagesimal fractions is shown also in a clay tablet described by A. Ungnad,^[12] In it the diagonal of a rectangle whose sides are 40 and 10 is computed by the approximation 40+2×40×10²÷60², yielding 42(13)(20), and also by the approximation 40+10²÷{2×40} yielding 41(15). Translated into the decimal scale, the first answer is 42.22+, the second is 41.25, the true value being 41.23+. These computations are difficult to explain, except on the assumption that they involve sexagesimal fractions.

15. From what has been said it appears that the Babylonians had ideograms which, transliterated, are Igi-Gal for “denominator” or “division,” and Lal for “minus.” They had also ideograms which, transliterated, are Igi-Dua for “division,” and A-Du and Ara for “times,” as in Ara–1 18, for “1×18=18,” Ara–2 36 for “2×18=36”; the Ara was used also in “squaring,” as in 3 Ara 3 9 for “3×3=9.” They had the ideogram Ba–Di–E for “cubing,” as in 27-E 3 Ba-Di-E for “3³=27”; also Ib-Di for “square,” as in 9-E 3 Ib-Di for “3²=9.” The sign A–An rendered numbers “distributive.”^[13]

 

1. His first papers appeared in Göttingische Gelehrte Anzeigen (1802), Stück 149 und 178; ibid. (1803), Stück 60 und 17.
2. In the division of the year and of the day, the Babylonians used also the duodecimal plan.
3. G. A. Barton, Haverford Library Collection of Tablets, Part I (Philadelphia, 1905), Plate 3, HCL 17, obverse; see also Plates 20, 26, 34, 35. Allotte de la Fuye, "En-e-tar-zi patési de Lagaš," H. V. Hilprecht Anniversary Volume (Chicago, 1909), p. 128, 133.
4. "On the Assyrian Mythology," Transactions of the Royal Irish Academy. "Polite Literature," Vol. XXII, Part 6 (Dublin, 1855), p. 406, 407.
5. George Smith, North British Review (July, 1870), p. 332 n.; J. Oppert, Journal asiatique (August-September, 1872; October-November, 1874); J. Oppert, Étalon des mesures assyr. fixé par les textes cunéiformes (Paris, 1874); Sir H. Rawlinson and G. Smith, "The Cuneiform Inscriptions of Western Asia," Vol. IV: A Selection from the Miscellaneous Inscriptions of Assyria (London, 1875), Plate 40; R. Lepsius, "Die Babylonisch-Assyrischen Längenmaasse nach der Tafel von Senkereh," Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin (aus dem Jahre 1877 [Berlin, 1878], Philosophisch-historische Klasse), p. 105-44.
6. The Babylonian Expedition of the University of Pennsylvania. Series A: "Cuneiform Texts," Vol. XX, Part 1. Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur (Philadelphia, 1906), Plate 15, No. 25.
7. See François Thureau-Dangin, Recherches sur l’origine de l’écriture cunéiforme (Paris, 1898), Nos. 485–91, 509–13. See also G. A. Barton, Haverford College Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905), where the forms are somewhat different; also the Hilprecht Anniversary Volume (Chicago, 1909), p. 128 ff.
8. G. A. Barton, op. cit., Plate 3, obverse.
9. Franz Xaver Kugler, S.J., Die babylonische Mondrechnung (Freiburg im Breisgau, 1900), Plate IV, No. 99 (81–7–6), lower part.
10. Symbols for such fractions are reproduced also by Thureau-Dangin, op. cit., Nos. 481–84, 492–508, and by G. A. Barton, Haverford College Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905).
11. “A Mathematical Cuneiform Tablet,” American Journal of Semitic Languages and Literatires, Vol. XXXVI (1920), p. 249–57.
12. Orientalische Literaturzeitung (ed. Peise, 1916), Vol. XIX, p. 363–68. See also Bruno Meissner, Babylonien und Assyrien (Heidelberg, 1925), Vol. II, p. 393.
13. Hilprecht, op. cit., p. 23; Arno Poebel, Grundzüge der sumerischen Grammatik (Rostock, 1923), p. 115; B. Meissner, op. cit., p. 387–89.