theodosius, de diebus et noctibus · 2012. 5. 23. · theodosius, de diebus et noctibus 13 the text...

Theodosius, De diebus et noctibus

Paul Kunitzsch and Richard Lorch

Key Words: day (yawm; nahar), ecliptic, equator, hemisphere (visi-ble and invisible), horizon, meridian line, night, Qust.a b. Luqa, spher-ical astronomy (before Ptolemy), Sun (position of), tropics (summerand winter), Theodosius of Bithynia.

AbstractTheodosius’ De diebus et noctibus was one of several texts of pre-Ptolemaic spherical astronomy. Like many other scientific texts it wastranslated from Greek into Arabic in the ninth century. Qust.a b. Luqawas possibly the translator. In two “books” comprising altogether 33propositions, it describes the lengths of days and nights according tothe various positions of the Sun on the ecliptic with respect to thetropics. In this paper the full text of three propositions (I 1, II 20 and21) alone is given; of the other propositions only the “enunciations”are here edited. The Arabic text is followed by an English translationwith comments.

Introduction

The three extant works of Theodosius of Bithynia (fl. 100 BC)1 arethe Sphaerica, the De habitationibus and the De diebus et noctibus.Together with Euclid’s Phaenomena and Autolycus’ De sphaera quaemovetur and De ortibus et occasibus, they belong to a group of textsthat present Greek spherical astronomy. These and other texts are

1On Theodosius, see Fecht 1-11; Ziegler coll. 1930–1932.

Suhayl 10 (2011) pp. 9–46

10 P. Kunitzsch and R. Lorch

usually considered to form a loose collection of works called the “littleastronomy” (mikräs ��ronomoÔmenos), in contrast to Ptolemy’s“Great Collection” (in Arabic later called Almagest)2. The De diebuset noctibus is about the lengths of days and nights when the Sun isat various parts of the ecliptic3.

These three works by Theodosius have all been edited in Greek:the Sphaerica by Heiberg (1927) and recently by Claire Czinczen-heim (2000) and the De habitationibus and De diebus by R. Fecht(1927)4. Further, all three works were translated into Arabic, per-haps in the ninth century. And of these the first two, Sphaerica andDe habitationibus, were translated from Arabic into Latin by Gerardof Cremona in the twelfth century. The Arabic and Latin versions ofSphaerica have been edited by Kunitzsch – Lorch 2010, and the Dehabitationibus by Kunitzsch – Lorch 2011. As for De diebus, whichwas not translated into Latin in the Middle Ages5, it was plannedto edit its Arabic version. Due to circumstances and lack of time, wehad to shorten the project. So we here edit only three of the thirty-three propositions of the text completely, i.e. with the mathematicalproofs, whereas of the other propositions only the enunciations arehere presented, so that the reader can see which problem is treatedin each proposition. In this way he can gain an impression of thecontents of the whole book.

Fecht edited the Greek text from only four manuscripts, the oldestof which MS Vat. gr. 204 of the 10th century. In her edition of Sphaer-ica Czinczenheim collated and fully described 23 Greek manuscripts,of which 15 also contain De habitationibus and De diebus, amongthem the four manuscripts used by Fecht. Several more manuscriptsof these two texts are listed in the data-base Pinakes: Textes et

2Cf. Ziegler col. 1932; Bulmer-Thomas 320a; Kunitzsch – Lorch 2011, 7.3For the contents of the work, see Heath 1921, II 246.4For some information on Fecht, see Kunitzsch – Lorch 2011, 8f.5An outline of De diebus was included by Dasypodius (Konrad Rauchfuß,

1530/32 – 1600/01) in his Latin work Sphaericae doctrinae propositiones, Stras-burg 1572 (cf. Sarton I 211; for Dasypodius, see Verdonk). In 1591 G. Auria (d.1615) published in Rome an annotated Latin translation “(libri duo) nunc pri-mum de Graeca in Latinam conversi” (this quotation is copied from the titlepageof the book, reproduced on the internet). On the defects of Auria’s translation,see Fecht 12.

Theodosius, De diebus et noctibus 11

manuscrits grecs, issued by the Institut de recherche et d’histoiredes textes, Paris6.

The Arabic version of De diebus differs from the version of Dehabitationibus in so far as, while the Arabic text of De habitationibusfollows the wording of the Greek text very closely, also in the useof the diagrams and the diagram letters, in De diebus only the textof the enunciations follows the Greek closely, but the text of themathematical proofs, the forms of the diagrams and the use of thediagram letters are mostly quite different from the Greek text (withvery few exceptions, which will be mentioned in the footnotes to theEnglish translation). In order to be quite sure of the authenticityof Fecht’s Greek edition, we ordered and inspected a copy of MSVat. gr. 204 on CD from the Vatican Library and compared it withFecht’s edition of 1927. We found that Fecht rendered everything –text, diagrams and diagram letters7 – exactly as it appears in MS2048. Therefore it seems that the difference between the text of themathematical proofs in the Arabic version and Fecht’s text eitheris derived from some variant of the Greek text or is due to someArabic revision of a translation closer to Fecht’s Greek text. We maynote that some diagrams – e.g. that of the very first proposition –show a difference in Greek and Arabic that cannot have come aboutby simply copying without reference to the text. Another differencebetween the Greek and Arabic versions of De diebus is manifest inthe numbering of the propositions in the two texts:

Arabic Greek

Book I Def. (3) (5 – two are absent in Arabic)Prop. 1 1

2 2 (+ Porisma, Fecht 68,8–18)

6Cf. Kunitzsch – Lorch 2011, 9 n. 11. We are grateful to Menso Folkerts forsearching this data-base.

7Also Fecht’s doubtful letters ,A and ,B (cf. Kunitzsch – Lorch 2011, 12f.) trulyrender what is in MS 204.

8Our doubts about the authenticity of Fecht’s edition expressed in Kunitzsch– Lorch 2011, 9 and 12f. are now disposed of. Neugebauer’s strictures (p. 752) onFecht’s treatment of the diagram of II 13 were based on the comparison of Fecht’sedition with MS Vat. gr. 191 (13c.), which we do not have at hand.


3 3 (+ Lemma 4, 72,25–74,5)4 4 (+ Porisma 80,19–82,8)5 5

6 77 68 89 10

10 911 1112 12

Book II Prop. 1 12 23 34 4 (Assumpta 1–4, 108,32–114,4, not in Arabic)5 56 67 78 8

Arabic Prop. 9 begins with “quasi lemma” (120,3–15) and enunciationof 9 (120,17–18). The text of Prop. 9 itself is a mixture of 120,19ff.and the lemma 122,19 – 124,7.There is no clear correspondence between Arabic Props. 10, 12 and13 and the Greek.

11 11 (cf. notes 25 and 26 to the English translation)14 1315 1416 (cf. note 29 to the English translation)17 1518 1619 1720 1821 19


The text of the Arabic translation edited below appears in twomanuscripts9:

A: Istanbul, Seray, Ahmet III 3464, ff. 124v–151v. The text is herewritten in three different hands: 124v–133v in the hand that haswritten most of the texts in the codex (ca. 1228 AD); a second handon 134r–149v; the third hand on 150r–151v. The third hand givesa date in the colophon: salkh shahr al-h. aram Rajab sanat thalathınwa-sittimi↩a hijrıya (about 12 May 1233)10.

K: Private library (formerly in the possession of H. P. Kraus), ff.135v–157r11. According to a notice on the fly-leaf, the manuscriptis in the hand of the famous North-African astronomer Abu ↪Alı al-Marrakushı, who was active in Egypt in the late 13th century. Char-acteristics of the writing suggest that the annotator is right12.

As it appears, the translator of De diebus into Arabic was Qust.aibn Luqa (d. ca. 300H/912-13 AD). His name is mentioned as transla-tor in A at the beginning of the text (124v) and in K at the beginningand end of Book I (135v and 145r) and at the beginning of Book II(145v). Neither the bio-bibliographers13 nor al-T. usı in his tah. rır ofDe diebus mention the name of a translator14.

The Arabic text below is followed by an English translation withcomments in the footnotes. For the diagram letters we there use theletters introduced by Gerard of Cremona in his Latin translationsof Sphaerica and De habitationibus15. As said above, we only edit

9A third manuscript of interest here (Lahore, private library of M. Nabı Khan)contains on pp. 185–294 the Arabic versions of Sphaerica and De habitationibus(cf. Kunitzsch – Lorch 2010, 3f.; Kunitzsch – Lorch 2011, 10). The late Dr. AntonHeinen very kindly gave us paper copies of these pages for our edition of the twotexts. It might be that the codex, on analogy with other majmu↪as of the kind,also contains the text of De diebus. Unfortunately, we no longer have access tothis library and therefore cannot ascertain that.

10For a detailed list of the 17 items contained in the manuscript, cf. Lorch 2001,22–23

11For a detailed list of contents of the manuscript, see Lorch 2001, 28.12Cf. Kunitzsch – Lorch 2011, 10f.13Ibn al-Nadım 269,5–7; Ibn al-Qift.ı 108,1–5 and 11–14.14Al-T. usı 1358, last text in the volume, p. 2; al-T. usı 1383, p. 147.15Cf. Kunitzsch – Lorch 2010, 8; Kunitzsch – Lorch 2011, 13.


the full text of I 1 and II 20–21. Therefore in the following table weshow only the letters occurring in these three propositions. In somepropositions, e.g. Fecht’s II 10, 11, 13 and 14, there occur letters orsymbols beyond W (cf. the tables of identities cited in footnote 15),which, however, do not appear in the three propositions edited herein full. Because, as said above, in the Arabic version of De diebusthe forms of the diagrams and the use of their letters are mostlydifferent from those in the Greek, we do not give in the followingtable Greek-Arabic equivalents of the diagram letters. Here followsthe table:

Arabic English

@ AH. Bk. GX Dè EP Z

h H

T¼ KÈ LÐ Mà N� S¨ Q

¬ F


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Yëð ] ùëð

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@

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Ðà

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[Fig. 2]

à Ð �ñ

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English Translation

[basmala etc.]

Book of Theodosius on Day [nahar ] and Night

Translation of Qust.a ibn Luqa al-Ba↪labakkı. It is two “books”

Book I

The Sun moves in an even motion contrary to the motion of theuniverse in the circle [through] the middle of the signs; this circle iscalled the sun-circle.The length of a day [nahar ] is the time from the rising of the Sun toits setting; the length of a night is the time from the setting of theSun to its rising1.The length of the rotation of the universe is the time in which each ofthe fixed stars moves from rising to rising or from setting to settingor from any position, whichever it is, to that position.

1 When the Sun travels [away] from the summer tropic and thenorth pole is above the horizon, a preceding day [yawm] is longer thana succeeding day and a succeeding night is longer than a precedingnight. When the Sun travels [away] from the winter tropic, it is thecontrary of that.

Example: we assume circle ABGD as some horizon in the world, thesummer tropic as circle AD and the circle of the signs as BHG. Letthe Sun travel [away] from the summer tropic, and let it rise on someday at point K ; on that day it traverses arc KL and sets at point L.So the length of the day [nahar ] is the time in which the Sun traversesarc KL. Also, let the Sun rise after that day at point M. I say: theday [nahar ] on which the Sun traverses arc KL is longer than the dayon which the Sun rises at point M.

Proof: we assume arc MN equal to arc KL. So, since the Sun tra-verses arc KL and arc MN in equal times – that is because it has

1Two definitions following here in the Greek text (Fecht, p. 54,7-16) are notrendered in the Arabic translation.


been assumed to move in an even motion – and2 when the Sun tra-verses arc KL, arc KL “cuts”3 the visible hemisphere. So, in the

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A

B westeast G

D

•••••••HKLMQSN

[Fig. 1]

time in which the Sun traverses arc MN, arc KL “cuts” the visiblehemisphere*. So, in equal times the Sun traverses arc MN and arc KL“cuts” the visible hemisphere. Arc KL “cuts” the visible hemispherein a time greater than the time in which arc MN “cuts” it4. That isbecause arc KL is nearer the touching-point of the summer tropic. Sothe Sun traverses arc MN in a time greater than the time in whicharc MN “cuts” the visible hemisphere. So, in the time in which arcMN “cuts” the visible hemisphere, the Sun traverses an arc smallerthan arc MN ; let us assume it traversed arc MS. So, when point N isin the west, the Sun – being at point S – will have set; point S sets

2and . . . hemisphere*, in K: and the Sun traverses arc KL in the time in whicharc KL “cuts” the visible hemisphere.

3The Greek text here uses the verb âxall��ein, as also Euclid, Phaenom-ena, ed. Menge – Heiberg, Definitions, p. 10,3–10 and ch. 14. In Euclid this wastranslated by Menge as permutare, a term later also used by Fecht in his Latintranslation of De diebus. The Arabic translation and al-T. usı in his tah. rır of Dediebus use the verb qat.a ↪a, which literally means “to cut” (which is also adoptedin the present English translation). For this terminology, see Neugebauer 759f.(where he cites in note 1 Nokk’s rendering “durchwandern” in his German trans-lation of the Phaenomena of 1850). For a mathematical explanation of the matter,see Neugebauer 765–767.

4Euclid, Phaenomena 14.


before point N, because it rises before it. In order that the Sun is seensetting, it must traverse an arc smaller than arc MS ; let us assume ittraverses arc MQ. So the time of day [nahar ] is the time in which theSun traverses arc MQ. Since arc KL is greater than arc MQ, the day[nahar ] in which the Sun traverses arc KL is greater than the day inwhich the Sun traverses arc MQ.

I say: a preceding night is shorter than a succeeding night. Example:we draw a second figure5 containing all that we have mentioned.

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A

B westeast G

D

•••••••HKLMNSQ

[Fig. 2]

We assume the Sun setting on some day at point K, and after itssetting at point K we assume it rising at point L. So the length of thenight is the time in which the Sun traverses arc KL. Also, we assumethe Sun setting on another day at point M ; let arc MN be equal toarc KL. I say: the night in which the Sun sets at point K is shorterthan the night in which the Sun sets at point M. Proof: since arcKL is equal to arc MN, the Sun traverses arc KL and arc MN in equaltimes. But in the time in which the Sun traverses arc KL, arc KL cutsthe invisible hemisphere. So in the time in which the Sun traversesarc MN, arc KL cuts the invisible hemisphere. So in equal times the

5It should be noted that in the proof for the second figure the diagram letters inGreek and Arabic are identical; the diagrams themselves, however, have a differentappearance.


Sun traverses arc MN and arc KL cuts the invisible hemisphere. ArcKL cuts the invisible hemisphere in a time shorter than the time inwhich arc MN cuts it. So the Sun traverses arc MN in a time shorterthan the time in which arc MN cuts the invisible hemisphere. So,in the time in which arc MN cuts the invisible hemisphere, the Suntraverses an arc greater than arc MN ; let us assume it traversed arcMS 6. So, when point N is in the east, the Sun – being at point S– has not risen, because point N rises before point S. In order thatthe Sun is seen rising, it must traverse an arc greater than arc MS ;let us assume that it traverses arc MQ. So the length of the night isthe time in which the Sun traverses arc MQ. Since arc KL is smallerthan arc MQ, the night in which the Sun traverses arc KL is shorterthan the night in which the Sun traverses arc MQ. So, when the Suntravels [away] from the summer tropic, a preceding day is longer thana succeeding day and a preceding night is shorter than a succeedingnight.

Similarly we explain that when the Sun travels [away] from the wintertropic, the contrary of that happens. Q.E.D.

2 When the Sun rises and sets on some day, its distance fromthe point of contact of one of the two tropics, whichever it is, beingequal, it reaches the point of the tropic in the middle of that day atthe meridian line. If that is the summer tropic, the day on which theSun reaches it is the longest day of the year. And the days and nightsin which the Sun travels from winter tropic to summer tropic andfrom summer tropic to winter tropic and of which the distance fromthe day in which the Sun reaches the point of the tropic is equal, areequal7.

3 When the Sun rises on some day at some point of one of theparallel circles, before reaching the point of the summer tropic, and

6MS is transmitted in the two Arabic manuscripts and in al-T. usı’s tah. rır ofDe diebus. The edited Greek text has NX (p. 58, l. 26), which we assume is aprinting mistake.

7The porisma added in the Greek text after the end of Prop. 2 (Fecht, p.68,8–18) follows in the Arabic text in Prop. 2 at the end, concluded by the usualformula for Q.E.D.


[when it] sets on some other day at a point of that circle, having leftthe point of the summer tropic, those two days [yawm] are equal.And the days and nights preceding one of these two days, the Suntravelling from winter tropic to summer tropic, and the days andnights that come after the other day, the Sun travelling from summertropic to winter tropic, those of them having the same distance fromeach of those two days are equal8.

4 When the Sun rises and sets on some day, its distance from oneof the two points of the tropics not being equal, it will not reach thepoint of the tropic in the middle of that day. The day on which theSun reaches the point of the tropic, if it reaches on it the summertropic, will be longer than all the days of the year, whose beginningis when the Sun reaches the winter tropic. The days that are near[the time] when the Sun reaches the semicircle nearer to the pointof the summer tropic, are longer than the days that are near [thetime] when the Sun reaches the other semicircle; and the nights areshorter, the contrary of what we have said. If it reaches the point ofthe winter tropic, the contrary of that happens9.

5 When the Sun travels [away] from the summer tropic and riseson the circle of the equator, the night before this rising is equal tothe day [yawm] on which it rises on the equator.

6 When10 the Sun sets and rises on two points opposite each otheror [when] there is between its rising and setting half a year, the nightis equal to the day [yawm].

7 The11 days and nights having equal distance from the equatorare equal. It is said that the distance of the days and nights from the

8The lemma for Prop. 4, added after the end of Prop. 3 in the Greek text (Fecht,p. 72,25 – 74,5), is added in the Arabic in Prop. 3 at the end, also concluded bythe formula for Q.E.D. It should especially be noted that the diagram letters inthis lemma are identical in Greek and Arabic.

9The porisma added after Prop. 4 in the Greek text (Fecht, p. 80,19 – 82,8) isincluded, in the Arabic text, in Prop. 4 at the end, concluded by the formula forQ.E.D.

10This proposition corresponds to Prop. 7 of the Greek text.11This proposition corresponds to Prop. 6 of the Greek text.


equator is equal when the distance of the rising-point is equal to thedistance of the rising-point and the distance of the setting-point isequal to the distance of the setting-point.

8 When the Sun traverses the semicircle embraced by the circleof the equator with the point of the summer tropic, the shortest daywill be greater than the longest night12.

9 When13 the Sun is retreating from the summer tropic, havingtwo settings, one higher and the other lower, its rising that followsits higher setting is above its rising that follows its lower setting.And14 its rising that precedes its higher setting is above its risingthat precedes its lower setting.

10 When15 the Sun travels [away] from the winter tropic andtwo risings are assumed for it, one higher and the other lower, thesetting that follows the higher rising is above the setting that followsthe lower rising. And16 the setting that precedes the higher rising isabove the setting that precedes the lower rising.

11 When the Sun travels [away] from the summer tropic, if ithas neither rising nor setting on the circle of the equator, there is noequality [istiwa ↩] of night and day [nahar ].

12 When the Sun is travelling [away] from the winter tropic, if ithas neither setting nor rising on the line of the equator, there is noevenness [i ↪tidal ], i.e. equality [istiwa ↩], of night and day [nahar ].

12The diagram letters in this proposition are identical in Greek and Arabic, butthe figures have a different appearance.

13This proposition corresponds to Prop. 10 of the Greek text.14This sentence, which is in Fecht’s edition of the Greek, p. 92,2–3, and in A,

is omitted in K.15This proposition corresponds to Prop. 9 of the Greek text.16This sentence, which is in Fecht’s edition of the Greek, p. 88,29–31, and in A,

is omitted in K.


Book II

1 When the Sun traverses the quadrant of the circle of the zodiacthat follows the beginning of Cancer, day and night together aredifferent from day and night together: a preceding day and night aregreater than a succeeding day and night together.

2 When the Sun traverses the quadrant of the circle [of the zo-diac] that follows [the beginning of] Libra, day and night together aredifferent from day and night together: a preceding day and night aresmaller than a succeeding day and night.

3 When the Sun traverses the quadrant of the circle [of the zodiac]that follows the beginning of Capricorn, day and night together aredifferent from day and night together: a preceding day and nighttogether are greater than a succeeding day and night together.

4 When the Sun traverses the quadrant of the circle [of the zo-diac] that follows the beginning of Aries, day and night together aredifferent from day and night together: a preceding day and night aresmaller than a succeeding day and night17.

5 Day and night together that follow the summer tropic are greaterthan day and night that follow the winter tropic; in the same way dayand night together are greater than the day and night together thatare their opposites.

6 Day and night together that follow the summer tropic are equalto day and night together that follow the winter tropic; in the sameway day and night together are equal to the day and night togetherthat are their opposites.

7 The days and nights whose distance from the equator on bothsides is equal are equal to the days and nights whose distance from theequator is equal when the Sun traverses the hemisphere that followsthe beginning of Cancer or that follows the beginning of Capricorn.

17After Prop. 4 follow in the Greek text (Fecht, p. 108,32 – 114,4) four “As-sumpta in sequentes propositiones” which do not appear in the Arabic translation.


8 The days and nights together whose distance from one of thetwo points of the tropics, whichever point it is, is equal are equal tothe days and nights that have that distance.

9 When18 the Sun reaches a solstice, whichever it is, in the middleof the night or the day [nahar ], the day [yawm] and night togetherthat are in the hemisphere have the same distance from the [point of]contact of the tropic at which the Sun has reached the solstice in themiddle of the day or night. Therefore19 it is clear that the daysand nights together that have an [equal] distance from the [point of]contact at which the Sun has reached the solstice in the middle ofthe day or night are equal to the nights and days that have the samedistance from it, just as their distance is on [. . .20] and on the majorityof the days and nights. At21 the other [point of] contact they arenot necessarily equal on the majority of the days and nights; moreovertheir distance will be equal in the distances of the straight lines only.If the Sun reaches a solstice, whichever it is, in the middle of the day[nahar ], it arrives at the meridian line equally in the middle of theday22. That23 is because it has become clear in Book I that, whenthe Sun reaches the solstice, the distance of [the points of] rising andsetting from the point of contact of the solstice is equal, i.e. that [thepoints of] rising and setting are on the same circle from [among] theparallel circles, because, when the Sun rises and sets on the samecircle from [among] the parallel circles, it reaches the solstice at themeridian line in the middle of the day, whichever solstice it is. Fromthat it is clear that, when the Sun reaches the solstice in the middle ofthe day, whichever solstice it is, it is in the middle of the day equallyon the meridian line. In that way it has also become clear that it [sc.the Sun], when it reaches the solstice, whichever solstice it is, in the

18The Greek text has added at the end of Prop. 8 a piece of text (Fecht, p. 120,3-15) called by Fecht a “quasi lemma”. In the Arabic version this text appears, atleast in part, as the enunciation for its Prop. 9. This beginning corresponds toFecht, p. 120,3ff.

19The following section corresponds to Fecht, p. 120,9ff.20Unidentified word in both Arabic manuscripts.21The following section corresponds to Fecht, p. 120,13ff.22This sentence corresponds to the enunciation of Prop. 9 in Fecht, p. 120,17–18.23For the beginning of what follows cf. Scholion 78 (Fecht, p. 170).


middle of the night, it is in the middle of the night at the meridiancircle. What is clear in the hemisphere that is below the Earth is[also] clear in the hemisphere that is above the Earth: the Sun hasabsolutely no other motion in the middle of the day and in the middleof the night equally at the meridian line. But, when it travels [away]from the summer tropic, it is in the middle of the day and the middleof the night between east and the meridian line in the hemisphere ofits course. And when the Sun travels [away] from the winter tropic, itis at the time of the middle of the day and the night equally betweenwest and the meridian line in the hemisphere of its course, the poleof the horizon being between the circles of the two tropics and thecircles touched by the horizon.

10 I say24 that the Sun reaches the middles of night in the place[in] the middle between the eastern hemisphere and the meridian line.

11 And I say that, when the Sun travels [away] from the wintertropic, it reaches the middles of day and night in the place that isbetween the western hemisphere and the circle of the meridian.

12 And I say that the Sun arrives in the middle of the night atthe place that is between the western semicircle and the circle of themeridian.

13 When25 the Sun reaches the tropic, whichever it is, in themiddle of the day or night, it reaches it at the meridian line. Thathas become clear, and it has also become clear that the Sun does notreach the meridian line on any other day or any other night, neitherin the middle of the day nor in the middle of the night. We wantto explain that on the days and nights in which the Sun reaches thetropic neither in the middle of the day nor in the middle of the night,it does not reach the meridian line, neither in the middle of the daynor in the middle of the night. Let26 us assume that the Sun reaches

24For this proposition and Props. 12 and 13 we find no close correspondence inthe Greek text.

25This portion of text corresponds to the passage appended to the Greek Prop.11, p. 132,12–19.

26The following lines show some resemblance to the enunciation of Prop. 12 inthe Greek text, p. 132,21–26.


the summer tropic at the moment of its rising. So I say that the Sunin the middle of the day is in the place that is between the easternhemisphere and the circle of the meridian.

14 Let27 the Sun reach the tropic before the middle of the day. SoI say: in the middle of the day the Sun will be between the easternhemisphere and the meridian line.

15 Also28, let the Sun reach the tropic after the middle of the day.So I say: in the middle of the day the Sun will be in the place that isbetween the western semicircle and the circle of the meridian.

16 In the same way29 we shall explain that the Sun, if it reachesthe point of the tropic before the middle of the day, it will be in themiddle of the day in the place that is between the meridian line andthe eastern hemisphere. If it reaches the point of the tropic after themiddle of the night, it will be in the middle of the night at the placethat is between half the western horizon and the meridian line. Atthe winter tropics, when the Sun reaches the tropic in the way thatwe have described, the contrary of that will happen. The proof for allthat is like the proof for it [i.e. what has been mentioned before].

17 When30 the year consists of complete rotations of the Sun,the days and nights in each year are equal to the days and nights ineach year in magnitudes [al-a ↪z. am] and number [al-kathra] and therising[s] and setting[s] on the horizon and on the circle of the Sun[i.e. the zodiac] are on the self-same points and the Sun reaches thepoints of the tropics and the equator line at the same hours, withoutdeviation.

18 But31 if the year does not consist of complete rotations of theSun, but the complete rotations are followed by a part of a rotation,

27This proposition corresponds to Prop. 13 of the Greek text.28This proposition corresponds to Prop. 14 of the Greek text.29This “proposition” partly corresponds to the last paragraph of the Greek

Prop. 14, p. 144,9–13.30This proposition corresponds to Prop. 15 of the Greek text.31This proposition corresponds to Prop. 16 of the Greek text.


the days and nights in a first year are not equal to the days and nightsin a second [succeeding] year and the rising[s] and setting[s] on thehorizon and on the circle of the Sun are not on the same point andthe Sun does not reach the two tropics and the equator at the samehour.

19 If32 we assume the times of the rotations of the Sun as equal,as appears to the sense, and we assume the year as [consisting of]complete rotations made by the Sun, then everything that we havedescribed goes in the same way every year, without deviation fromwhat we have said earlier. If the year does not [consist of] completerotations of the Sun, but the complete rotations are followed by apart of a rotation, then, if the remaining part is commensurable toone rotation, what we have described does not happen in the yearsfollowing the first year, but rather it follows after some years. So, af-ter a number of years that will happen. That becomes clear throughthe following procedure: we take two numbers in mutual relationto one another in the ratio of a [complete] rotation to the part of therotation exceeding the complete rotations. We look at the amount ofthe greater number, and in a corresponding [number] of years all thatwe have described returns to the first situation. If the part remainingafter the complete rotations is not commensurable to a complete ro-tation, the matter never returns to the same situation. In the opinionof Callippus, who maintains that a year [consists of] 365 days and 1

4of a day, the return happens every four years.

20 In33 the opinion of Meton and Euctemon and those who main-tain that a year [consists of] 365 days and 5

19 of a day, the rotationsreturn to their [original] situation in nineteen years.Example: we draw again the first [i.e. preceding, in Prop. 19] figureand assume the Sun rising at point E. Let it rotate 365 rotations andend up at H. The remaining arc EH it traverses in 5

19 of a day. I say:in nineteen years all the rotations return to their [first] situation34.

32This proposition corresponds to Prop. 17 of the Greek text.33This proposition corresponds to Prop. 18 of the Greek text.34In this proposition the letters H – L in the argumentation and the diagram are

identical in Greek and Arabic. The four letters M – O (Greek) = M – Q (Arabic)indicating the small portions of the ecliptic are different: in the Greek they runfrom right, M near E, to the left, whereas in the Arabic they begin with M at theleft and end with Q (= Greek O) on the right, near E. The demonstrations for


Proof: we assume each of arcs HT TK KL equal to arc HE. We dividearc EH into five nineteenths at points M N S Q, and we assume an

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A

B G

D

•E•H

.....

Q.....

S

.....

N.....

M

•T

•K

•L .....

F

arc like arc FL35 to be 119 of a rotation. Then it becomes clear that

the Sun in the second year starts its course at point H, completes365 rotations and reaches point T. In the third year it reaches pointK, and in the fourth year point L; it traverses arc LQ, a [complete]rotation, and reaches point Q. So, in the four years, when the Sunmakes the complete rotations, it misses an arc like arc QE, whichis 1

19 of a rotation. So, in sixteen years of [complete] rotations theSun misses arc ME, which is 4

19 of a [complete] rotation. In the threeremaining years, missing every year five nineteenths of a rotation, itcompletes arc ML, which is 15

19 of a rotation, and traverses arc FE,which is the rotation, and ends up at point E, where it had started.

these letters are consistent within each of the two traditions, but different betweenGreek and Arabic. A special situation is found near the left end of the arc of theecliptic near L. Here the Greek text has the letter rho, R (with the omission of pi,P, which normally follows in the series of letters after O). The Arabic translation,instead, has F (rendering Greek P), as if the Arabic translator really saw P in hisGreek source; if it had been rho, R, that would have been rendered in the Arabic

system by�

� (q), which would appear in Gerard’s Latin translations and ourEnglish version as C. In the Arabic diagrams, in K the letter F is omitted; in A

it is there, but miswritten as ð (waw).35For F, cf. the preceding footnote.


So, in nineteen years the rotations return to their [first] situation.Q.E.D.

21 If36 the exceeding fraction is not commensurable to a rotation,the rotations never return to [the point of] their first start. Thatbecomes clear through the following operation.

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A

B G

D

E•H

We redraw the figure preceding this [proposition] and we assume thatthe Sun has risen at point E. It has rotated 365 times and has endedup at H. There remains arc HE, but arc HE is not commensurable toa rotation37. I say: the rotations will never return to their [first]situation.

Proof: if it were possible that the Sun rise in some year at point E,then, missing every year an arc equal to arc HE and reaching in theremaining year point E, it traverses the remaining arcs in completerotations. The remaining arcs are “measured”38 by a rotation; arc

36This proposition corresponds to Prop. 19 of the Greek text.37The use of the letters E and H in this proposition is identical in Greek and

Arabic.38The Arabic text here uses the verb ↪adda, ya↪uddu, which originally means

“to count”, “to number”; it stands for the Greek verb metrèw(Fecht, p. 154,23and 24). This reminds us of Euclid, Elements VII, Def. 3-5 and 8-14, where forsimilar mathematical operations the verb katametrèw was used (transl. Heath, “tomeasure”; German transl. Thaer, “messen”). The Arabic translation of Ish. aq ibn


HE may also “measure” them. So arc HE is commensurable to therotation; but it had been assumed incommensurable. Therefore theSun has never a rising at point E, and the rotations never return totheir [first] situation. Q.E.D.

[A] Finished is Book II of Theodosius’ Book on Days and Nights. Godbless his prophet Muh.ammad and his family. At the end of the holymonth of Rajab in the year of the Hijra 360 [= ca. 12 May 1233].

[K] Finished is Book II of Theodosius’ Book on Night and Day. Withits end the entire book is finished. Praise be to God, the Lord of theworlds. God bless Muh. ammad, his family and his companions andgive [them] peace.

Bibliography

Berggren: J. L. Berggren, “The Relation of Greek Spherics to EarlyGreek Astronomy”, Sources and Studies in the History and Phi-losophy of Classical Science 2 (1991) 227–248.

Bulmer-Thomas: Ivor Bulmer-Thomas, “Theodosius of Bithynia”,Dictionary of Scientific Biography XIII (1976) 319a–321a.

Busard 1983a: H. L. L. Busard (ed.), The First Latin Translationof Euclid’s Elements Commonly Ascribed to Adelard of Bath,Toronto: Pontifical Institute of Mediaeval Studies, 1983.

H. unayn revised by Thabit ibn Qurra renders that also by the verb ↪adda, ya↪uddu(De Young), which, in turn, appears in the Latin translation of Adelhard of Bathas numerare (Busard 1983a, p. 196), in the Latin translation of Gerard of Cremonaas numerare (Busard 1983b, col. 165 – but in Def. 4, instead, metiri for the sameArabic ↪adda), and in the so-called Adelard II version also as numerare (Busard–Folkerts, I, p. 187 – but in Def. 10 [corresponding to 12 in Ish. aq and Gerard] and12 metiri). It may well be that both Theodosius and his Arabic translator herefollow the terminology in Euclid VII, Def. 3ff.


Busard 1983b: H.L. L. Busard (ed.), The Latin translation of the Ara-bic version of Euclid’s Elements commonly ascribed to Gerard ofCremona, Leyden: New Rhine Publishers, 1983.

Busard–Folkerts: H. L. L. Busard and M. Folkerts (eds.), Robert ofChester’s (?) Redaction of Euclid’s Elements , the so-called Ade-lard II Version, 2 vols., Basel – Boston – Berlin 1992.

Czinczenheim −→ Theodosius, Sphaerica

De Young: G. R. De Young, The Arithmetic Books of Euclid’s El-ements in the Arabic Tradition: an Edition, Translation, andCommentry, Unpublished thesis, Harvard University, Cambridge,Massachusetts, 1981.

Euclid, Elements: C. Thaer, Euklid, Die Elemente, Buch I – XIII,Darmstadt, 81991.

Euclid’s Phaenomena: Euclidis opera omnia, ed. J. L. Heiberg et H.Menge, vol. 8, Leipzig 1916.

Euclid’s Phaenomena: Euclid’s Phaenomene, ubersetzt und erlautertvon A. Nokk. Beilage zum Freiburger Lyceums-Programm von1850.

Fecht −→ Theodosii De hab.

Heath: Sir Th. L. Heath, The Thirteen Books of Euclid’s Elements,Translated from the Text of Heiberg, 2nd ed., 3 vols., New York:Dover Publications, s. d. (first published in 1956).

Heath 1921: Sir Th. L. Heath, A History of Greek Mathematics, 2vols., Oxford 1921.

Heiberg −→ Euclid, Phaenomena; Theodosius, Sphaerica

Ibn al-Nadım, Kitab al-Fihrist, ed. G. Flugel, J. Roediger and A.Muller, 2 vols., Leipzig 1871–1872; repr. Beirut: Maktabat Khay-yat., s.d.

Ibn al-Qift.ı, Ta↩rıh˘

al-h. ukama ↩, ed. A. Muller and J. Lippert, Leipzig

1903; repr. Baghdad: Maktabat al-Muthanna, and Cairo: Mu↩as-sasat al-Khanjı, s.d.

Kunitzsch – Lorch 2010 −→ Theodosius, Sphaerica


Kunitzsch – Lorch 2011 −→ Theodosius, De hab.

Lorch 2001 −→ Thabit

O. Neugebauer, A History of Ancient Mathematical Astronomy, 3vols., Berlin – Heidelberg – New York 1975

Nokk −→ Euclid, Phaenomena

Sarton: G. Sarton, Introduction to the History of Science, vol. I, Bal-timore 1927.

Sezgin: F. Sezgin, Geschichte des arabischen Schrifttums, vol. V:Mathematik bis ca. 430 H., Leiden 1974; vol. VI: Astronomiebis ca. 430 H., Leiden 1978.

Thabit ibn Qurra, On the Sector-Figure and Related Texts, ed. R.Lorch, Frankfurt am Main: Institut fur Geschichte der Arabisch-Islamischen Wissenschaften 2001; repr. Augsburg: E. RaunerVerlag (Algorismus, 67) 2008.

Thaer −→ Euclid, Elements

Theodosii de habitationibus liber, de diebus et noctibus libri duo, ed.R. Fecht (Abhandlungen der Gesellschaft der Wissenschaften zuGottingen, Phil.-hist. Kl., N.F. XIX,4), Berlin 1927; repr. Nen-deln: Kraus, 1970.

Theodosius, De habitationibus, Arabic and Medieval Latin Transla-tions, ed. P. Kunitzsch and R. Lorch (Bayerische Akademie derWissenschaften, Phil.-hist. Klasse, Sitzungsberichte, 2011, Heft1), Munchen 2011.

Theodosius Tripolites, Sphaerica, ed. J. L. Heiberg (Abhandlungender Gesellschaft der Wissenschaften zu Gottingen, Phil.-hist. Kl.,N.F. XIX,3), Berlin 1927.

Theodosius, Sphaerica: Claire Czinczenheim, Edition, traduction etcommentaire des Spheriques de Theodose, these (Universite ParisIV–Sorbonne), 2 vols., Lille: Atelier national de reproduction destheses 2000.

Theodosius, Sphaerica. Arabic and Medieval Latin Translations, ed.P. Kunitzsch and R. Lorch (Boethius, 62), Stuttgart 2010.


Al-T. usı, Nas.ır al-Dın, Majmu ↪ al-rasa ↩il, Hyderabad 1358 [/1939].

Al-T. usı, Nas.ır al-Dın, Tah. rır-e mutawassit.at, Tehran 1383/2005 (Fac-simile edition of MS 3484 in the Public Library of Tabriz).

Verdonk: J. J. Verdonk, “Dasypodius, Cunradus”, Dictionary of Sci-entific Biography III (1971) 585a–b.

K. Ziegler, “Theodosios [5]”, Paulys Real-Encyclopadie der classi-schen Altertumswissenschaften, 2. Reihe (R–Z), vol. V.A.2 (Stutt-gart 1934), col. 1930–1935.

theodosius, de diebus et noctibus · 2012. 5. 23. · theodosius, de diebus et noctibus 13 the text...

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