Bisecting the trapezoid: tracing the origins of a Babylonian computation of Jupiter’s motion

Abstract

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics (1900–1700 BCE). Some implications of the use of geometric methods by Babylonian astronomers are also explored.

Appendix

This “Appendix” contains commented transliterations of selected mathematical texts that were discussed above and of the four astronomical trapezoid procedures. Only selected issues that are relevant in connection with the astronomical trapezoid procedures are addressed here; for complete commentaries the reader is referred to the editions and studies that are mentioned below. Note that for practical reasons the letter khet is represented as a plain h instead of one with a little arc below it. A line number such as 6a denotes the first part of line 6, 6b the rest of that line.

1.1 UET 5, 858

This Old Babylonian tablet from Ur was first edited by Vaiman (1961, 250–254) and thereafter by Friberg (2000), who followed Vaiman’s interpretation. For a copy of the cuneiform text see Friberg (2000). The line numbering used here follows that of I. Vaiman. The text is written completely in Sumerian, the only apparent exception being the originally Akkadian conjunction ${u}_{\mathrm{3}}$, “and”, which is here a loan word in Sumerian (Edzard 2003, 161–162). In accordance with the conventions for transliterating Sumerian texts, all signs are transliterated in small letters, except if the Sumerian reading is unclear.

This transliteration follows I. Vaiman, except for some updated sign readings (murub$_{\mathrm{4}}$, DU). Multiplication is expressed by x a.ra$_{\mathrm{2}}$ y.še$_{\mathrm{3}}$ DU, literally “to go x times y” (7, 13, 18, 21, 27), which is rare in Old Babylonian problem texts, but common as a logogram in Late Babylonian mathematics and astronomy. The correct Sumerian reading of DU cannot be addressed here. Forms with the prospective prefix u$_{\mathrm{3}}$ have been tentatively translated as imperatives (Edzard 2003, 122) rather than as temporal clauses “when/after ...”, because the former is more appropriate for instructional texts. This is unproblematic for u$_{\mathrm{3}}$.ub.te.zi (11, 16, 24), which can be interpreted as a 2nd person form, but it is problematic for u$_{\mathrm{3}}$.ub.gar (5, 20, 25) and u$_{\mathrm{3}}$.ub.DU, which are not 2nd p. forms according to standard Sumerian grammar.⁶²

Obv. 3: 15 (error for 10?): one expects 10, the mean width of the second partial trapezoid, $(d+W_2)/2$, because the preceding 15 represents the mean width of the first partial trapezoid, $(W_1+d)/2$.

5: 10 sag, “head”: this word has been ignored in the previous editions. One might expect the name of the figure (sag.ki gu$_{\mathrm{4}}$ = trapezoid) to be written here. The intended meaning remains unclear.

8, 22, 28: 1(iku) iku: here 1(iku) denotes the numeral 1 with the wedge written horizontally, which is used for expressing areas in iku. In all three instances this wedge is also (assumed to be) followed by the sign iku itself (the latter is transliterated as aša$_{\mathrm{5}}$ in Friberg 2000).

1.2 TMS 26: §3 and part of §4

This Old Babylonian problem tablet originates from Susa (Iran). Only §3 and the beginning of §4 are covered here. The transliteration follows the edition by Muroi (2001), which incorporates significant new insights. For the first edition and the cuneiform text see Bruins and Rutten (1961, 124–129).

A notable feature of this tablet, thus far not found in other mathematical texts about bisected trapezoids, is that the verb ${\textit{z}}\hat{a}{\textit{zu}}$, “to divide”, (side I 13; side II 4, 12, 14, 16) is written with the logogram BAR. This verb also occurs in YBC 4675 (see below), where it is written phonetically. This reading of the ambiguous logogram BAR is implied by the writing BAR-zu in II 14 (see Muroi 2001). Hence ${\textit{z}}\hat{a}{\textit{zu}}$ could also be the intended Akkadian reading of BAR in Texts B and D (see below). Squaring is expressed both with A.RA$_{\mathrm{2}}$, “times” (I 14), which is the usual term for multiplication in the Late Babylonian period, and GU$_{\mathrm{7}}$.GU$_{\mathrm{7}}$, literally “to make eat itself” (II 7, 8), the most common term for squaring in Old Babylonian mathematics.

Side II 4: zu-uz = ${\textit{z}}\bar{u}{\textit{z}}$, “divide” (imp.): the use of ${\textit{z}}\hat{a}{\textit{zu}}$ (BAR) for division elsewhere in the text suggests this restoration instead of i-di-in, “give” (imp.), proposed in Muroi (2001).

7, 8: GU$_{\mathrm{7}}$.GU$_{\mathrm{7}}$ = $\check{s}{\textit{ut}}\bar{a}{\textit{kil}}$, “make eat itself”. This form, which appears phonetically as šu-ta-ki-il in VAT 8512 (see main text) and in other tablets (for examples see Friberg and Al-Rawi 2016), is an imp. m. sg. of $\check{s}{\textit{ut}}\bar{a}{\textit{kulum}}$, a Št stem of ${\textit{ak}}\bar{a}{\textit{lum}}$, “to eat”. The logogram GU$_{\mathrm{7}}$.GU$_{\mathrm{7}}$ is a reduplicated form of the Sumerian verb GU$_{\mathrm{7}}$, which also means “to eat”. Like the alternative verb $\check{s}{\textit{utamhurum}}$(NIGIN), “to make confront one another”, $\check{s}ut\bar{a}{\textit{kulum}}$(GU$_{\mathrm{7}}$.GU$_{\mathrm{7}}$) denotes the construction of a square from its side and, simultaneously, the computation of its area (Høyrup 2002, 23–25). According to an alternative interpretation proposed by Høyrup (2002, 23), this form is to be analyzed as $\check{s}{\textit{utak}}\bar{u}{\textit{lum}}$, “to make hold one another”, in which case the use of GU$_{\mathrm{7}}$.GU$_{\mathrm{7}}$ could reflect a pun, but Friberg and George (2010: 124) have proven that $\check{s}{\textit{ut}}\bar{a}{\textit{kulum}}$, “to make eat itself”, is the correct interpretation.

9: 2.53.20.DA IB$_{\mathrm{2}}$.[SI$_{\mathrm{8}}$.BI ka-bi-is 1.12]: “With 2,53,20 [pace off its square] root, [it is 1,12]”: see Muroi (2001) for a discussion of this expression, which appears to denote a procedure for approximating the square root, where kabis, “pace off” is an imp. of ${\textit{kab}}\bar{a}{\textit{su}}$.

13: li-zu-zu = ${\textit{liz}}\bar{u}{\textit{z}}\bar{u}$, “let (the brothers) divide”. This restoration by K. Muroi is prompted by the use of ${\textit{z}}\hat{a}{\textit{zu}}$(BAR) for division elsewhere in the text.

1.3 AO 17264: selected passages

This tablet came to light in unscientific excavations, presumably in Uruk. It has been dated to the Kassite era (1400–1200 BCE). For editions see Thureau-Dangin (1938, 74–76), Neugebauer (1935, 126–134), Thureau-Dangin (1934. For other investigations see Friberg (2007b, 292–295), Brack-Bernsen and Schmidt (1990, 4–25), Vaiman (1961, 204–206), Gandz (1948, 22–27).

Obv. 1: : no convincing interpretation has been suggested for these signs (Thureau-Dangin 1938, 74–76; Neugebauer 1935, 126–134). One would expect the name of the figure, a quadrilateral or trapezoid, to be mentioned here.

2, 3: NIGIN$_{\mathrm{2}}$: the context requires that this means “to be equal”. The intended Akkadian reading is unclear, because this meaning of NIGIN$_{\mathrm{2}}$ is not attested elsewhere.

3: ${\textit{lim}}\hat{a}{\textit{tu}}$, “perimeters”: this literal translation replaces “area”, which was adopted as an ad-hoc translation in all previous editions. A meaning “area” is not attested for this word and no areas are actually computed in the text.

13, 14: UR.KA.E: the context requires a meaning “to square”, but this logogram is otherwise not attested as far as known. The intended Akkadian reading is unclear.

15: U.BI.GAR: this formally prospective form is most likely to be read as an Akkadian imperative or 2nd p. present tense of ${\textit{kam}}\bar{a}{\textit{ru}}$, “to add” (literally “to accumulate”).

30: mu-ut: abbreviation of muttarrittum, “descending line”.

1.4 YBC 4675 obv. 1–16

This Old Babylonian problem text kept in the Yale Babylonian Collection originates from unscientific excavations, perhaps in Larsa. For editions of the tablets see Høyrup (2002, 244–249), Neugebauer (1945, 44–48). For a new translation by Robson (2007, 101). Further investigations of the tablet are Friberg (2007b, 272–274) and Friberg (1990, 562).

1, UŠ UŠ I.GU$_{\mathrm{7}}$, “(one) length is inclined to (the other) length”: Here I.GU$_{\mathrm{7}}$ stands for ikkal, literally “it eats”, which is used for inclined sides in Babylonian mathematics.

3, ${{\textit{zu-u}}_{\mathrm{2}}{\textit{-uz}}}$ = ${\textit{z}}\bar{u}{\textit{z}}$, “divide”: for the use of this verb in bisection problems see the commentary to AO 17264.

1.5 Erm 15073 §3

This Old Babylonian tablet kept in the Hermitage (St. Petersburg) originates from unscientific excavations, perhaps in Sippar (Vaiman 1961, 232; Friberg 2007b, 304). For the first edition by Vaiman (1961, 232–244); for a partial edition by Friberg and a photograph of the tablet see Friberg (2007b, 304–308). The same column numbers and line numbers are used here as in Vaiman 1961. Friberg has exchanged obverse and reverse and renumbered the columns accordingly.

Obv. iii 10: AN.TA GAR.RA, “put it on top”: this refers to the act of writing down the number 0;0,40 (= f) for subsequent use (line 12).

13: wa-ri-tam is an accusative of ${{\textit{w}}{\bar{a}}{\textit{rittum}}}$, “that which descends”, which here denotes the partial length. Recall that in AO 17264 the cognate word muttarrittu, “descending line”, is used in the same sense.

1.6 Text B = BM 34757 P1$^\prime$

This fragment arrived in the British Museum around 1879, having been excavated unscientifically in Babylon. Apart from the left edge, no other edges of the original tablet are preserved. The textual restorations imply that about 2 cm of clay are missing from the right side. The present edition incorporates some new insights, but the transliteration is identical to that in Ossendrijver (2016), Supplementary Materials, 5–7. For a photograph of Text E see Fig. S1 in the latter publication. For an edition of the complete tablet see Ossendrijver (2012, No. 38).

Side Y 1, ${{\textit{ina }}\check{s}{\textit{u-tam-hi}}}$- , “by squaring (?)”: see the discussion of this term in the main text.

2, UŠ-$\check{s}{\textit{u}}_{\mathrm{2}}$, “its length”: for a new interpretation of this term see the main text.

7, : the context requires that this term, tentatively translated “crossing”, denotes the position where Jupiter has reached half the distance of 10;45 UŠ. For a discussion of this word, which is not attested elsewhere in mathematical or astronomical texts, see Ossendrijver (2016).

7, SE$_{\mathrm{3}}$: as argued in Ossendrijver (2016), two readings of this logogram appear possible here. First, a form of ${{\textit{ma}}\check{s}\bar{a}{\textit{lu}}}$, “to be half; equal”, most likely a stative of the G stem, ${\textit{ma}}\check{s}{\textit{il}}$, “it is halved”. With this reading the preceding ina libbi(ŠA$_{\mathrm{3}}$)-$\check{s}u_{\mathrm{2}}$ is most suitably interpreted with an instrumental meaning, “whereby”. Alternatively SE$_{\mathrm{3}}$ can be interpreted as a second person of the present tense of the G stem of $\check{s}{\textit{ak}}\bar{a}{\textit{nu}}$, “to place”, i.e., ${\textit{ta}}\check{s}{\textit{akkan}}$, “you place”, in which case ina libbi(ŠA$_{\mathrm{3}}$)-$\check{s}u_{\mathrm{2}}$ can be translated “within it; in its middle”.

9, BAR: This logogram expresses the halving. While it was previously thought to represent a form of ${\textit{ma}}\check{s}\bar{a}{\textit{lu}}$, “to be half; equal” (Ossendrijver 2016), the Old Babylonian problem text TMS 26, discussed above, suggests that it could also be a form of ${\textit{z}}\hat{a}{\textit{zu}}$, “to divide”, presumable a stative m. sg. ${\textit{z}}\bar{u}{\textit{z}}$, “divided”. The same logogram occurs in Text D (see below).

12, , “of 10;45”: the meaning of this incompletely preserved phrase remains clear. It presumably belongs to a descriptive term qualifying the factor 0;30 (=1/2) by which the outcome of line 11 is multiplied here. Traces of several numerals, perhaps forming the number 9.30 (= 0;9,30?) could not be interpreted.

1.7 Text C = BM 34081 $+$ 34622 $+$ 34846 $+$ 42816 $+$ 45851 $+$ 46135 P5$^\prime$

The six extant fragments of the tablet BM 34081+ 34622+ 34846+ 42816+ 45851+ 46135 arrived in the British Museum between 1878 and 1881, having been excavated unscientifically in Babylon. The left and right edges of the original tablet are partly preserved, but nothing remains of the original upper and lower edges. On both sides the tablet is divided into three columns. The present transliteration is identical to the one in Ossendrijver (2016), Supplementary Materials, 7–8. For a photograph see the latter publication, Fig. S3; for an edition and photograph of the complete tablet see Ossendrijver (2012, No. 18).

Obv. i 23, ṭe-ri-tu$_{\mathrm{2}}$, “pinched”: in Text B this word appears in an alternative name for the small width of the trapezoid, but there is insufficient context for determining its meaning here.

1.8 Text D = BM 35915 P1$^\prime$

The tablet BM 35915 arrived in the British Museum around 1880, having been excavated unscientifically in Babylon. No edges of the original tablet are preserved. Text D occupies one entire side of the fragment; the other side is destroyed. It was first edited in Ossendrijver (2016), Supplementary Materials, 8–10. For a photograph of the tablet see the latter publication. The present transliteration is identical to that in Ossendrijver (2016).

The badly damaged lines 1–2 can be assigned to Part I. They can be translated as follows:

(several lines missing) $^{\text{ side } \text{ X } \text{1 }}$[...] ... [...] $^{\text{2 }}$[...] [10];45 [you add] to the position of appea[rance, and ...]

All that can be understood is the instruction to add 10;45 to Jupiter’s position at its first appearance.

4, BAR: as mentioned in the commentary to Text B, the Old Babylonian problem text TMS 26 (see above) suggests that this logogram could also represent a form of ${\textit{z}}\hat{a}{\textit{zu}}$, “to divide”, in addition to a form of ${\textit{ma}}\check{s}\bar{{\textit{a}}}{\textit{lu}}$, “to be half; equal”.

1.9 Text E = BM 82824 $+$ 99697 $+$ 99742 P1$^\prime$

This tablet consists of three fragments which arrived in the British Museum in 1883 and 1884, having been excavated unscientifically in Babylon. No portion of the original edges of the tablet is preserved. Only one side is inscribed; the other side is destroyed. Textual restorations in the other procedures imply that not much clay is missing from the left side, perhaps a few cm from the right side. The present edition incorporates a few corrections to the previous edition (Ossendrijver 2016, Supplementary Materials, 10–13). For a photograph of Text E see Fig. S4 in the latter publication; for an edition and photograph of the complete tablet see Ossendrijver (2012, No. 40).

Lines 1–4a are tentatively assigned to Part I, but it is unclear where exactly Part I ends. Several measures of the trapezoid are mentioned, but these lines continue to defy interpretation. The following translation incorporates a few corrections to Ossendrijver (2016).

(unknown number of lines missing) $^{\text{ side } \text{ X } \text{1 }}$[...] ... [...] $^{\text{2 }}$[...] length$^{\text{? }}$ ... [...] ... the length, 0;12$^{\text{? }}$, the la[rge] width, [...] $^{\text{3 }}$[...] ... the small width, ... .. [...] 30$^{\text{? }}$ was carried ... [...] $^{\text{4a }}$[...] from the days ...

Side X 2, UŠ$^{\text{? }}$, “length$^{\text{? }}$”: erroneously translated as “width” in Ossendrijver (2016).

3: xxx: these signs could not be identified. The reading suggested in Ossendrijver (2016), IGI 2.50?, is probably incorrect and does not allow a meaningful interpretation.

5: 27$^{\text{? }}$: as argued in Ossendrijver (2016), this must be an error for 28. The following lines imply that the result is passed on as 28, but this number looks like 27. The phrase “there remains” occurs almost exclusively after subtractions, which strongly suggests that 28 (days) was obtained by subtracting 32 days, the number partly preserved at the end of line 4, from 60 days, which can be restored in the following gap.