This project on the editions of Euclid’s Elementa is dedicated to the memory of two illustrious researchers and good friends, namely

Note: We are grateful to Myungsunn Ryu for his edition of the Greek Elementa ΣΤΟΙΧΕΙΑ ΕΥΚΛΕΙΔΟΥ and his generous copyright policy, in accordance with which we have used his images for our edition, if not otherwise noted. 

A COMPARATIVE ANALYSIS OF FOUR MEDIEVAL LATIN TRANSLATIONS OF EUCLID’S ELEMENTS

The Elements (or Στοιχεῖα) is one of the most read and important mathematical works in the history of Science. Having been unavailable to Western Europe for more than a millennium, the twelfth century witnessed a wave of translations of this work into Latin. In this paper I will analyze four twelfth century Latin translations of Elements by Euclid, three of which were translated from Arabic, and one, which I will pay special attention to, from Greek.  I intend to start this paper with a short presentation of the life and works of the four different translators, in a – as far as possible – chronological order. In addition, I will take a closer look at the problems that occur in the Sicilian translation of the text, since this translation is based on the technique known as ad verbum de verbo (word by word). Lastly, I will compare and discuss the way the four different translators apply different methods to achieve the translation process, concentrating on the first book of this work.During this process my main aim will be to reveal the variations found in the different texts.

The four translations

As mentioned above, I intend to focus on four different translations of the Euclidian text. These texts provide an interesting case study as they represent a new tradition of translating and reading mathematics in the twelfth century. Under the next four headings I will provide a brief life overview of the men who translated the four texts I will be working with; important due to the link that can be found in the ability to access manuscripts among the translators working from Arabic. I would therefore like to briefly introduce the texts they had the ability to reference when making their translation. The first Arabic translations of Elements were made by al-Hajjaj ibn Yusuf ibn Matar, who provided the Arabic world with two translations dating to the end of the 8^th /beginning of the 9^th century[1]. The next translation was made by Ishaq ibn Hunayn in the tenth century[2], and was revised by Tabit ibn Qurra, who died in 901[3]. Bearing this connection, I will now turn my attention toward the lives of the translators.

Adelard of Bath

Adelard of Bath was born in England in the late eleventh century, and became one of the leading English medieval scholars. He flourished in the time between 1116 and 1142, but before that he traveled for some time in Europe, starting in France where he studied at Tours, and ending up in Bath again around 1130, after visiting “Salerno, Sicily (…), Cilicia, Syria and possibly Palestine.[4]” Adelard of Bath provides us with the earliest Latin translation of Euclid’s Elements known to us, dated to approximately 1130 AD[5]. In all, three versions are ascribed to him, of which the first is probably based on the al-Hajjaj II version[6]. Of these three translations only the first two are translations, while the third is a commentary[7].

Gerard of Cremona

Not much is known about Gerard of Cremona until he moved to Toledo where he learned Arabic in order to translate works from this language into Latin, finding that in Toledo there were many significant works written in Arabic, that the Latin language was lacking[8]. His translation of the Elements was based upon the Arabic translation ascribed to Ishaq ibn Hunayn from the tenth century[9]. Busard further claims that “it is very likely that Gerard did not utilize texts of both making his translation, but based his labors on an Ishaq-Thabit text that already contained material drawn from al-Hajjaj I.[10]” Gerard was one of the most productive translators of his time, and during the time he spent in Toledo he translated 71 works from Arabic into Latin[11]. He lived and worked in Toledo until his death in 1187.

Hermann of Carinthia

Hermann of Carinthia lived in the twelfth century AD, and was active a bit later than Adelard of Bath[12]. Even though we do not know too much about his life, we know that the first work he translated into Latin was the De revolutionibus mundi or Prognostica by Sahl Ben Bishr, and that he did this in the year 1138, and that he only wrote one independent work, known as De essentiis[13]. He lived and worked in the Ebro valley, and belonged to a school of intellectuals who worrshiped classical Latin, and most of all the Latin of Cicero, which one clearly can see when reading his translations. In the process of making his translation of Elements from Arabic to Latin it is – according to Busard – likely that he used the Arabic translation ascribed to al-Hajjaj[14]. This because of the Arabism found in his text.

The anonymous translator from Sicily

The last translator is the anonymous translator from – most likely – Sicily, whose translation of the Elements is the only known translation of this work from Greek to Latin (all the others are made from Arabic). We don’t know much about this translator, but according to John E. Murdoch he published his translation of Elements in the last half of the twelfth century[15]. We also know that he most likely was the translator of the Almagest by Ptolemy (also from Greek to Latin)[16].

There is, however, uncertainty as to whether this translator translated the whole work, or if the books XII and XIII were translated by another writer. Murdoch[17] claims that the former is the case, while Busard[18] thinks the latter is the case. I will, on my hand, not enter this discussion since I primarily want to focus on the first book of Elements.

Ad verbum de verbo

In the translation made in Sicily by the anonymous translator it is clear that it is the verbum de verbo technique that is applied. To cover this technique briefly, I intend to give some examples from the text to show how this form of translation works, in addition to addressing problems that our translator would have to overcome.

The first example is taken from proposition I.1, and shows in basic how this technique works in practice:

            Ἐπὶτῆςδοθείσηςεὐθείαςπεπερασμένηςτρίγωνονἰσόπλευρονσυστήσασθαι[19].

            Super datam rectam terminatam trigonum isopleurum constituere[20].

We see how every single word of the Latin text corresponds with the equivalent Greek word from the original: ἐπὶwith super, δοθείσηςwith datam, εὐθείαςwith rectam, πεπερασμένηςwith terminatam, τρίγωνονwith trigonum, ἰσόπλευρονwith isopleurum, and finally συστήσασθαιwith constituere. The only different being that the Greek τῆςδοθείσηςεὐθείαςπεπερασμένηςis dative, due to the preposition ἐπὶ, whilst the Latin equivalent datam rectam terminatam is accusative due to the preposition super. The obvious reason for this is that, despite translating word by word, one must still bear in mind the languages’ different rules and different cases ruled by different prepositions.

One problem that occurs when trying to translate word by word directly from Greek to Latin is that both languages possess features that are nowhere to be found in the other; for instance the usage of articles, that are present in Greek, but do not exist in Latin. This fact forces our Sicilian translator to use other forms in Latin. Let us look at some examples of this: In proposition I.2 Euclid writes “δεῖδὴπρὸςτῷΑσημείῳτῇδοθείσῃεὐθείᾳτῇΒΓἴσηνεὐθεῖανθέσθαι.[21]”, and our translator translates it to “Oportet ergo a puncto a recte b g equam rectam ponere.[22]” Here we can see that he is not trying to reproduce the Greek articles in his Latin text, probably because the article is not needed in order to tell the case or gender of neither the point a, nor the line b g, since the noun puncto gives away both gender and case of a, and the adjective (here used as a noun) recte gives away the gender and case of b g. In other propositions, however, he tries to reproduce the articles by applying a pronominal followed by a relative clause. An example of this we find in proposition I.36: “λέγω, ὅτιἴσονἐστὶτὸΑΒΓΔπαραλληλόγραμμοντῷΕΖΗΘ.[23]” A sentence that our translator translates as follows: “Dico quoniam equale est a b g d ei quod est z i t e.[24]” As mentioned above, he reproduces the Greek article τῷwith the Latin pronominal ei and the relative clause quod est. He seems to do this when there neither exist a noun nor adjective that give away the letters’ case (in this case z i t e): Since the case is needed in order to understand the meaning of the sentence and articles are nowhere to be found in the Latin language, our translator has to use the case of a pronominal. In this case the pronominal is, ea, id is used to reproduce the Greek article; and because one cannot put this pronominal in the same way as the Greek article, one must use a relative clause, in this case quod est.

            Another interesting feature about this translation occurs in proposition I.44 among others. Our translator often transcribes some of the Greek words into Latin, rather than translate them, occasionally taking this practice to the extreme, as seen in the proposition mentioned above when he translates the following sentence “Παρὰτὴνδοθεῖσανεὐθεῖαντῷδοθέντιτριγώνῳἴσονπαραλληλόγραμμονπαραβαλεῖνἐντῇδοθείσῃγωνίᾳεὐθυγράμμῳ.[25]” into “Penes datam rectam dato trigono equale parallilogrammum paraballein in dato angulo rectilineo.[26]”. Here we see both the normal transcription (παραλληλόγραμμονto parallilogrammum) where he has borrowed the Greek word, but still uses the nominative ending found in the Latin declination system, and a more extreme version (παραβαλεῖνto paraballein). In the latter case we see that he keeps the Greek word as well as its Greek infinitive ending, instead of trying to fit it into the Latin declination system as he usually tries to do. This he also does in proposition I.45, again with the same verb, only this time declined in the perfect future imperative middle/passive third person singular; the Greek sentence “καὶπαραβεβλήσθωπαρὰτὴνΗΘεὐθεῖαντῷΔΒΓτριγώνῳἴσονπαραλληλόγραμμοντὸΗΜἐντῇὑπὸΗΘΜγωνίᾳ, ἥἐστινἴσητῇΕ.[27]” is now translated into “Et parabeblistho penes i t rectam trigono d b g equale parallilogrammum i m in angulo i t m qui est equalis angulo e.[28]” These two cases provide us with some interesting examples for comparison with the rest of his translation of the first book of the Elements: In most other cases he translates the Greek verb into Latin, even the places where the verb βαλεῖνoccurs; as is the case in the second postulate of the first book. Here Euclid writes “Καὶπεπερασμένηνεὐθεῖανκατὰτὸσυνεχὲςἐπ᾽εὐθείαςἐκβαλεῖν.[29]” and the anonymous translator translates it to “Et terminatam rectam in directo secundum continuum emittere.[30]” Notice the verb: Instead of transcribing it to the Latin case – which is the case with paraballein – he translates the verb ἐκβαλεῖνapplying the Latin verb emittere. Worth noticing is the fact that he does not always translate ἐκβαλεῖνwith emittere, in fact he applies the verb educere in most cases, as he does in proposition I.29, where he translates the Greek original “αἱδὲἀπ᾽ἐλασσόνωνἢδύοὀρθῶνἐκβαλλόμεναιεἰςἄπειρονσυμπίπτουσιν:[31]” with the Latin sentence “Recte vero a minoribus quam sint duo recti educte in infinitum concident.[32]” It is therefore tempting to draw the conclusion that the reason why he is translating παραβαλεῖνwith paraballein is the lack of a Latin prefix equivalent to the Greek prefix παρα.

Comparing the translations

Living in the same century, but in different parts of Europe using different Arabic (and in the Sicilian’s case Greek) manuscripts and different translation techniques makes these four translations an interesting object of study because of their relations in time and the differences in places. Despite my lack of knowledge considering the Arabic language, I still think it is interesting to see how they differs from each other, especially taking into consideration the time they were translated in and the state of the Latin language at that time. This was after the Latin language had ceased to exist a native language, but before the renaissance where intellectuals began to worship the classic ideal of the language once more. Thus we are in a “limbo” phase, during which people knew Latin without knowing it in its classical form. I will, during the next five headings analyze and discuss the four different translations, with a specific look at their usage of the accusative with infinitive versus the later equivalent constructed by using quia/quoniam/quod with indicative to construct oblique clauses. Under the fifth heading I will undergo a short comparison of the four translations looking at both orthography and structure.

The translation ascribed to Adelard of Bath

Being the first Latin translation of Elements known to us, Adelard of Bath’s translation is the best point to start.  One of the most striking features of this translation is his inconsistency concerning the construction of oblique clauses. This makes it very interesting to analyze and discuss the use of the accusative with infinitive construction versus the construction involving the verb of utterance (in these cases dico) withquia/quoniam/quod and then the verb for the oblique clause in indicative. Analyzing this matter, I will focus on the oblique clauses that depend on the verb dico, even though oblique clauses, in this translation, also follow other verbs, for example the periphrastic manifestum est found throughout the whole translation (an example of which, is to be found in proposition I.46 (I.47 in the Greek original)), where he writes “Manifestum igitur est quia quadratum ex ductu b g in seipsum existens est(…)[33]”. Here we see that he applies the construction quia + indicative, and not the accusative with infinitive. Unlike Gerard and our Sicilian translator, who – as we shall see below – rarely applies the construction dico + quia + indicative, Adelard of Bath’s favorite construction for oblique clauses depending on dico is – at least in the first book of the Elements – that very one; as seen already in proposition I.1, where he writes “Dico quia ecce fecimus triangulum equalium laterum supra lineam a b assignatam.[34]” In fact, he utilizes this construction 37 times in the first book of the Elements. He does so in the following propositions: I.1, I.2, I.3, I.4, I.5, I.6, I.13, I.14, I.16, I.17, I.18, I.19, I.20 (twice), I.25, I.26 (twice), I.27, I.28 (twice), I.29, I.30, I.31, I.32, I.34, I.36, I.37, I.38, I.39, I.40, I.41, I.42, I.43, I.45, I.46 and I.47. When it comes to applying quoniam + indicative when wanting to construct an oblique clause depending on dico, Adelard of Bath only does this once – in proposition I.24: “Dico itaque quoniam alkaida b g alkaida h z longior.[35]” also omitting the verbesse (which in this case would be est). He never applies the construction dico + quod + indicative. When it comes to the use of the accusative with infinitive, Adelard of Bath is obviously familiar with the construction, as seen in proposition I.8: “Dico angulum b a g angulo h d z equalem.[36]” (Also in this sentence he omits esse.) Despite this it is not his most preferred construction. In fact, he only utilize this construction seven times, in the following propositions: I.7, I.8, I.9, I.15, I.21, I.33 and I.35. This tells us that although he, in the first book, uses the construction first appearing in late antiquity/early middle ages, he is in fact familiar with the construction applied in classical Latin (in every case an oblique clause is depending on the verb dico) and has the ability to implement it.

            Another feature that is worth noticing in Adelard’s translation is his use of Arabism. Like our Sicilian translator who tends to use Greek words from time to time – as we saw was the case with the verbπαραβαλεῖν–Adelard of Bath does the same thing with Arabic words, but is not consistent in his usage of these words. A good example of this is his usage of the words alkaida and basis – meaning base. He utilizes these words when the text is talking about βάσις, but – as I said – he is not consistent when it comes to applying only one of the words. In fact he keeps on, throughout the whole first book, mixing alkaida and basis when translating the Greek word βάσις. Some examples of this can be found in proposition I.4 when he writes “Dico quia b g reilqua basis basi h z relique quam respicit equalis est.[37]”, differing from proposition I.5 when he writes “Suntque supra alkaidam b g. Sicque item demonstratum est sub alkaida duos angulos equales z b g et H g b suntque sub triangulo.[38]”

            Another detail of this translation is the numbering of the propositions. Euclid’s original text contains 48 propositions, consistent with the translations by the anonymous Sicilian translator and Gerard, but not this text: It only contains 47. This is because the propositions I.26 and I.27 – as they appear in the Greek original – have been joined together to form one proposition (proposition I.26). 

Gerard’s translation           

Because the construction of oblique clauses is a good way to see how much knowledge a translator would have about the Latin language, it is interesting to take a closer look at – as we did with the translation ascribed to Adelard of Bath – Gerard’s use of the accusative with infinitive contra quod/quia/quoniam with indicative. Gerard tends to prefer different constructions than Adelard. Let us look at some examples: In proposition I.4 he gives the following translation using the accusative with infinitive construction: “Dico igitur reliquum latus b g, quod est basis, reliquo lateri e z equale esse, quod est eius relativum.[39]” Here we see that the accusative with infinitive – Dico (…) reliquum latus (…) equale esse (…) – rather than the form from the medieval/late antiquity is applied. However, Gerard does not always apply this construction; in fact he also utilizes, from time to time, dico + quia/quoniam/quod + indicative. This could be due to one of two things; one being that Gerard did not have the competence of the writers from the Renaissance, that he is in fact unsure whether he should use the accusative with infinitive or the other construction; or two, that he really possess the same understanding of the Latin tongue as his predecessors, but in spite of this does not have the same idea of what is “good” Latin (classic Latin as it was in the golden age of Rome with Caesar and Cicero) and what is not (Latin from late antiquity and later), as the Renaissance man would have. An example of this can be found in proposition I.5 where he writes “Dico igitur quod angulus a b g est equalis angulo b g a et quod angulus g b d angulo b g e equatur.[40]”  This construction is one of the late antiquity/medieval type. Instead of the accusative with infinitive construction (with which the sentence would look something like this: dico igitur angulum (…) equari) he now writes Dico igitur quod angulus (…) est. Thus we see his variation concerning translations of oblique clauses; some places he uses accusative with infinitive, other quia/quod/quoniam with indicative. In fact, he applies the construction first occurring in the late antiquity 22 times, in which quia is applied only one time (that is in proposition I.1) , and quod 21 times (this happens in propositions I.3, (I.4), I.5, I.6, I.16, I.17, I.20, I.24, I.26, I.27, I.28, I.30, I.31, I.32, I.33, I.35, I.37, I.39, I.41, I.44,  and I.46) . The accusative with infinitive, on the other hand is taken in use 29 times in the first book of the Elements as it occurs in Gerard’s version. This takes place in the following propositions: I.2, I.4, I.8, I.9, I.10, I.11, I.12, I.13 (in this proposition he omits the verb esse), I.14, I.15, I.18, I.19, I.21, I.22, I.23, I.25, I.26, I.28, I.29, I.34, I.36, I.38, I.40, I.42, I.43, I.44, I.45, I.57, and I.48. It is interesting that Gerard, unlike the Greek original (and of course the translation from Sicily) ends every one of his exempli causa parts with a claim constructed by using dico and the oblique clause.

             Gerard has in his translations pieces that are nowhere to be found in the Greek original, due to two reasons; one that parts were added by the Arabic translators earlier in the Middle Ages, and two that he added sections himself. In the first book we find several additions, but two stand out as the most striking ones due to their significant lengths, and the fact that these additions are only to be found in Gerard; none of the other translators in this paper have these two additions, which is remarkable considering that they have all (except the anonymous translator from Sicily) used the same Arabic versions of Elements in their process of making Latin translations of this important work. These two examples can be found in propositions I.35 and I.44, of which I will now briefly discuss. Having ended proposition I.35 with the usual sentence “Et hoc est quod demonstrare voluimus.[41]”he adds the following paragraph, not to be found in the other translations made from Arabic:

Huius preterea theorematis probatio aliter invenitur disposita hoc modo, scilicet, quia linea az iam cecidit super duas equidistantes lineas, scilicet, zg; eb, provenit angulus extrinsecus aeb intrinseco sibi opposito ezg equalis. Latera quoque omnia superficierum equidistantium linearum que sibi opponuntur et anguli sunt equalia, unumquodque ei quod sibi opponitur equale, videlicet, latus zg lateri be et latus gb lateri ze et lateri da equale. Latus ergo ad lateri ez equale existit. Cetera que sequuntur non mutantur nisi quod dicitur: et quia duo latera gz et zd duobus lateribus be et ea sunt equalia et angulus gze equalis angulo bea, erit basis dg basi ab equalis etcetera.[42]

            One can, however, come to the proof to this theorem in another way when it is arranged like this: because the line az have fallen upon two straight lines, namely zg andeb, the exterior angle aeb, that is equal to the interior and opposite angle ezg, occurs. And because in every parallelogram all the sides and angles are equal to the ones that are opposite to it, every one [of the sides and angles] are equal to the ones that are opposite; that is, the side zg equal to the side bg, and the side gb equal to the sides ze and da. Thus the side ez is equal to the side ad. The other ones are not changed unless it is said so. And because the two sides gz and zd are equal to the two sides be and ea, and the angle gze is equal to bea, the base dg will be equal to the base ab etc.[43]

The purpose of adding this paragraph is apparently to clarify, as well as explain the conditions on which the solution of the theorem of the proposition is built. Euclid takes it for granted that the reader can agree with the conditions presented in the proposition without having to prove that they are in fact true. Gerard obviously thinks that this needs to be explained and proved in order to clarify the process of solving the theorem presented in the beginning of the proposition. It explains, in short, all the conditions that Euclid omits, even though one has to accept them in order to prove the theorem.

Later in the first book he adds another addition to proposition I.44, but now the purpose is altered a bit:

Huius quoque theorematis dispositio invenitur hoc modo: Sit linea data bk que producatur usque ad punctum a, et constituam super lineam ba superficiem equidistantium laterum triangulo gde equalem abns, et ponam angulum eius abs angulo z data equalem, et protraham lineam ns in rectitudine ad punctum m, et coniungam lineam bk puncto m producendo lineam equidistantem bs; an lineis, et coniungam punctum b puncto m protrahendo lineam bm. Linea autem km linee an equidistat et iam cecidit super eas nm linea. Ergo duo anguli anm; nmk duobus rectis angulis sunt equales. Duo igitur anguli anm; nmb duobus rectis sunt minores. Sed cum due recte linee producuntur a duobus angulis minoribus duobus rectis, impossibile est quin concurrant. Due igitur linee mb; na in rectitudine protrahantur et coniungantur in puncta l. Deinde producam a puncta l lineam rectam ad punctum t equidistantem duabus lineis ak; nm, et protraham lineam mk in rectitudine ad punctum t et lineam sb ad punctum h. Dico igitur quod iam constituimus super lineam bk superficiem equidistantium laterum triangulo gde equalem cuius angulus angulo z est equalis.

Probatio et cetera non mutantur.[44]

The disposition of this theorem is also to be found in this way: Let bk be the given straight line which shall be drawn all the way to the point a. Then I will construct, over/upon the line ba the parallelogram abns equal to the triangle gde, and I will make its angle abs equal to the given angle z. Then I will draw the line ns straight to the point m, and let the line bk be joined with the point m by drawing a line parallel with the lines bs and an. Then, let the point b be joined with the point m by drawing the line bm. But the line km is parallel with the line an and the line mn has fallen over them. Thus the two angles anm and nmk are equal to two right angles. Therefore the two angles anm and nmb are less than two right angles. But when two straight lines are drawn from two angles less than two straight angles, it is impossible that they don’t meet. Thus the two lines mb and na, when drawn straight, meets [and are joined] in the point l. Then I will draw from the point l a straight line to the point t parallel to the two lines ak and nm, and I will draw the line mk straight to the point t, and the line sb to the point h. I say that we already/now have constructed over the line bk a parallelogram equal to the triangle gde, and that one of its angles is equal to the angle z. The proof and the rest are not changed.

As I recently mentioned, he adds this paragraph, but not exactly for the same purpose as last time. Here he shows a different process, with which the problem can be solved, and the figure constructed; it is a different way of approaching the task, and a different way leading to the same result. Instead of doing it the Euclidian way, he simply wants to prove to us all that there exists a different way to solve the task.

Hermann of Carinthia’s translation

One of the first things that strikes you when reading Hermann’s translation of the Elements in comparison to those of Adelard and Gerard is the simplicity and lack of structure. Unlike Adelard and Gerard, Hermann seems to be lacking the clear and elaborated disposition found in the other two translations. Where both Adelard and Gerard apply exempli gratia (or other similar formulations) in order to introduce the construction phase/claim of each proposition, and rationis causa (or other equivalents) before the proof of each proposition, Hermann does not do that, but simply writes the whole proposition as one, without dividing it up into smaller pieces for the sake of making it clearer and more perspicuous.

            Unlike the other translators, Hermann rarely applies oblique clauses ruled by dico in order to postulate a claim in the end of the construction face, just before he commence to dissert the proof. Unlike Adelard and Gerard, who are fording a claim by saying dico and what they claim to have done before every proof, and the anonymous translator/Euclid (I say so because the Sicilian translation is in fact a Latin mirror to the Greek original) applies these sentences in front of the proofs of the propositions in which constructing is the main quest, Hermann does not use these summarized claims in front of many profs: in fact he only applies this construction in the following 8 propositions: I.2, I.4, I.5, I.18, I.23, I.34, I.42 and I.45. In this cases he applies the construction quod with indicative (“Dico ergo quod tetragonus (…) equalis est eis, quos g a et b a in se ducta conscribunt.[45]”)  three times (1.4, 1.23 and I.45) and the accusative with infinitive (“Dicimus itaque terminales superficies que sunt a z  et z d sibi invicem equales.[46]”) five times (I.2, I.5, I.18, I.34 and I.42). Irrelevant for the meaning, but worth commenting is the fact that he in some of these propositions – unlike all the other translators – applies the passive neutral perfect form of the verb dico. In the propositions I.5, I.28 and I.23 he writes dictum est instead of dico: “dictum est autem a z equalem esse a h et a b ei que est a g,[47]” 

A striking feature one discover as soon as one take a look at the first book of the Elements in Hermann’s translation, is that his first book only contains 46 propositions, as opposed to the original text, which contains 48. This is because Hermann combines proposition I.11 with I.12, and the proposition I.26 with I.27. His motivation for doing so might be that the propositions he joins together are quite similar to one another, describing almost the same.

The translation of the anonymous translator

As mentioned above, the Latin language is in a rather delicate state during this period in history, and a good example of this is the lacking of the accusative with infinitive, as well as problems with the declination system. Investigating the use of the accusative with infinitive versus the use of quod/quia/quoniam with indicative after a verb of utterance – dico being the most used verb of such in our texts – one soon discovers that the Sicilian translation does not contain any traces of the accusative with infinitive at all. One might wonder why it is so, but the answer is simple: Translating a Greek text, using the ad verbum de verbo method, one cannot use the accusative with infinitive to follow the verb dico, because this construction is not applied after the Greek verb λέγω; one simply have to use the construction from late antiquity, namely dico followed by quod/quia/quoniam and the verb for the oblique clause in indicative, since the Greek way of constructing an oblique clause depending on λέγωis to apply ὅτιand the verb in indicative. Let us look at some examples of this: In proposition I.4 Euclid writes “λέγω, ὅτικαὶβάσιςἡΒΓβάσειτῇΕΖἴσηἐστίν,[48]” and our Sicilian translator translates it to “Dico quoniam et basis b g basi e z equalis est[49]”. We see that our translator uses the dico quoniam with indicative, just as the Greek text does. Further exemplification is unnecessary because the anonymous translator choose this solution every time the Greek λέγωὅτιoccurs in the first book of the Elements. The propositions wereλέγωὅτιoccurs, and our translator translates it with dico quoniam, are the following: I.4, I.5, I.6, I.8, I.9, I.10, I.11, I.12, I.13, I.14, I.15, I.16, I.17, I.18, I.19, I.20, I.21, I.22, I.24, I.25, I.26 (twice), I.27, I.28, I.29, I.30, I.32, I.33, I.34, I.35, I.36, I.37, I.38, I.39 (twice), I.40 (twice), I.41, I.43, I.46, I.47, and I.48. This means that in all 42 cases were λέγω+ ὅτιis used in Greek, the anonymous translator uses dico + quoniam + indicative.

            He is also confused when it comes to the gender of some of the Latin nouns; an example of which, is the word punctum – a word that I will return to in under the heading “a short comparison of the four translations”. In proposition I.5 he translates the Greek “ΕἰλήφθωγὰρἐπὶτῆςΒΔτυχὸνσημεῖοντὸΖ,[50]” with the Latin phrase “Sumatur enim in recta b d quodlibet punctum sitque z[51]”. In this particular sentence he takes punctum as a neutral noun and thus also applies the neutral form of the noun quilibet/quaelibet/quodlibet.But in proposition nine he reproduce the Greek sentence “ΕἰλήφθωἐπὶτῆςΑΒτυχὸνσημεῖοντὸΔ,[52]” with the Latin “Sumatur in recta ab punctus quilibet sitque d.[53]” Here he obviously decline the noun punctus as it was a noun of the masculine gender, and “correctly” he also uses the masculine form of the pronoun quilibet. In fact he uses punctum/punctus as a masculine noun 21 times in the first book and as a neutral noun only six times. This is, however, an imperfect presentation of the usage of the two genders because the endings of the accusative, genitive, dative and ablative is the same in both genders, which means that whenever the word appears as an object for a clause or dependant on a preposition, the ending will be the same no matter which gender he thinks punctum/punctus is. An example of this is to be found in proposition I.4 when he translates the Greek sentence “ἘφαρμοζομένουγὰρτοῦΑΒΓτριγώνουἐπὶτὸΔΕΖτρίγωνονκαὶτιθεμένουτοῦμὲνΑσημείουἐπὶτὸΔσημεῖοντῆςδὲΑΒεὐθείαςἐπὶτὴνΔΕ, ἐφαρμόσεικαὶτὸΒσημεῖονἐπὶτὸΕδιὰτὸἴσηνεἶναιτὴνΑΒτῇΔΕ·[54]” with the Latin sentence “Coaptato enim a b g trigono super d e z trigonum et posito a quidem puncto super d punctum, recta vera a b super rectam d e, coaptabit et b punctus super punctum e eo quod equalis est recta a b recte d e.[55]” We see out of this example that he reproduce the Greek B σημεῖονwith the Latin b punctus and that this of course is a noun of the masculine gender; but the other places in which σημεῖον– now declined in different cases – is to be translated, it is in fact impossible to tell whether it is masculine or neutral (e.g. super punctum, and a quidem puncto – the word can, in these examples, be read as both neutral and masculine since the declination system is the same for neutral and masculine nouns in the second declension system, the only exception being nominative (and vocative) singular (-us (m), -um (n), -e (voc. m.)), and nominative and accusative plural (-i, -os (m.), -a, -a (n.))).

A short comparison of the four translations

An interesting detail that is hard not to notice reading these four translations is the confusion round the gender of the noun punctum/punctus. In classical Latin this word is neutral and is thus written as punctum, and this should also be the case in Medieval Latin, but obviously it is not. In the first definitionσημεῖόνis translated with punctus by Hermann, Adelard and the anonymous translator, but punctum by Gerard. The Greek original “Σημεῖόνἐστιν, οὗμέροςοὐθέν.[56]” is reproduced as the following Latin sentences: Hermann translates it (from Arabic) to “Punctus: est cui pars non est.[57]”, Adelard (also from Arabic) with “Punctus est illud cui pars non est.[58]”, the Sicilian translator (from Greek) into “Punctus est cuius pars nulla.[59]”, whilst Gerard on the other hand translates the sentence (from Arabic) into “Punctum est cui pars non est.[60]” An answer to why they are doing so (they are also using this word as both neutral and masculine within their very translations, as we showed earlier) can be that they take the word as a perfect participle of the verb pungo, and/or the state of the Latin language at the time.

            Interesting it is to look at the different translators solutions in term of disposition and structure in their translations (the anonymous translator follows Euclid completely, and is therefore left out of the discussion for now). The three translators translating from Arabic shows some variation concerning this matter. Gerard’s translation is very vell-structurtured indeed,  which is clearly llustrated in the way all the propositions are arranged: first the actual proposition (what it is that he is supposed to do), then follows the construction/instruction (how to make the figure that is to be constructed, or part of it explained), introduced by the phrase verbi gratia or exempli gratia, followed by a claim that says that he have done what he is supposed to, then follows the proof (proving that he is right), introduced by the words huius probatio (est) or probatio eius, before he ends the proposition repeating what it was to be done or proven. One will soon discover that Adelard follows the same procedures as Gerard and that the structure of his translation is very similar to that of the Greek original. Like Gerard he applies an introduction in order to announce his proof by writing rationis causa, and he does so in front of every proof in all the propositions. Concerning the introduction to the constructions, Adelard seems to apply a somewhat different solution than Gerard: He does not utilize such a solution in the same extent as Gerard.

Hermann on the other hand lacks the clear structure found in the two former translations, but still obtains the progression within each proposition as found in the Greek original text. By this I mean that though he does not divide the proposition in a clear way using headings as Gerard and Adelard, he still follows the Greek original the way it is preserved in the Arabic translations. This does not necessarily mean that he is less structured than Adelard and Gerard, but that it simply is a different way of arranging each proposition. In  order to show in practice how the four different translators applies different solutions in terms of arranging a proposition, I will present the first proposition as it appears in the different translations (not showing the Greek original I will let the Sicilian version represent it since this is a direct translation). The translations will be presented in the same order as they were discussed earlier in the paper, starting with Adelard, followed by Gerard, then Hermann, ending with the translation of the anonymous translator from Sicily:

Nunc demonstrandum est quomodo superficiem triangulam equalium laterum super lineam rectam assignate quantitatis faciamus.                                                                        Sit linea assignata a b. Ponaturque centrum super a occupando spacium quod est inter a et b circulo, supra quem g d b. Item ponatur supra centrum b occupando spacium inter a et b circulo alio, supra quem g a h. Exeantque de punto g supra quem incisio circulorum due linee recte ad punctum a et ad punctum b. Sintque ille g a et g b. Dico quia ecce fecimus triangulum equalium supra lineam a b assignatam.                        rationis causa: Quia punctum a factu est centrum circuli a d b, facta est linea a g equalis linee a b. Et quia punctum b est centrum circuli g a h, facta est linea b g equalis linee b a. Sicque unaqueque linearum g a et g b equalis est linee a b. Equalium autem uni rei unumquodque equale alteri. Itaque linee tres a g  et a b et b g invicem equales. Triangulus igitur equalium laterum a b g factus est supra lineam a b assignatam. Et hoc est quod in hac figura demonstrare intendimus.[61]

Now Gerard:

            [1.1] Super rectam lineam definite quantitatis triangulum equilaterum constituere.

Verbi gratia: Ponatur linea recta ab definite quantitatis, et super centrum a secundum quantitatem spatii quod est inter a et b, circumducarur circulus, super quem sunt g; d; b. Alius quoque circulus super centrum b secundum quantitatem spatii quod est inter a et b describatur super quem sunt g; a; e. Deinde a puncto g in quo unus duorum circulorum alium secuit,due recte linee ad duo pupcta a et b protrahantur sintque linee ga et gb. Dico igitur quia iam fecimus triangulum equilaterum super lineam ab datam.

Huius probatio est: Quia punctum a factum est centrum circuli gdb,fit linea ag equalis linee ab. Et similiter quia punctum b factum est centrum circuli gae, fit linea bg linee ba equalis. Unaqueque igitur harum duarum linearum ga; gb linee ab equalis invenitur. Que autem eidem rei equalia sunt, sibi quoque invicem sunt equalia. Tres igitur linee ag; ab et bg sibi invicem sunt equales. Triangulus igitur abg est equalium laterum qui, ut ostensum est, super lineam datam ab constitutus est. Et hoc est quod demonstrare voluimus.[62]

Then Hermann:

(I.1) Primum igitur equilaterum triangulum supra rectam et definite quantitatem lineam collocamus.

Data siquidem linea recta inter a et b puncta, acceptaque punctorum distancia, id est linee spacio fixo circino supra a centrum fiat circulus b g d, translato statim equale tenace, eodemque spacio retento, circa b centrum fiat alter circulus a g h. Deinde ab g sectionis circulorum puncto descendant recte linee in a et b. Eritque huiusmodi equilaterus triangulus. Nam a centrum circuli b g d, lineam a g ei que est a b equalem esse cogit. Simili quoque modo b centrum circuli a g h lineam b g ei que est b a coequatur. Centrum enim est punctus a quo omnes linee ad circulum exeuntes sibi invicem equales sunt. Est autem si duo uni sunt equalia, utrumque alteri esse equale. Quaniam ergo duo chateti basi sunt adequati, et sibi invicem sunt equales. Est itaque triangulus equilaterus supra rectam lineam collocatus.[63]

Then finally the translation of the anonymous translator:

{I.I} Super datam rectam terminatam trigonum isopleurum constituere. Esto data recta terminata a b. Oportet ergo super a b rectam trigonum isopleurum constituere.

Centro quidem a; diastimati vero a b circulus scribatur g d b. Et rursum centro b, spatio vero a b circulus scribatur g a e. Et a puncto g secundum quod secant se invicem circuli in a, b puncta copulentur recte g a et g b. Et quoniam a punctus centrum est circuli g d b, equalis est a g recte a b. Rursus quoniam b punctus centrum est circuli g a e, equalis est b g recte b a. Ostensa est autem et g a recte a b equalis. Utraque ergo rectarum g a et g b recte a b est equalis. Eidem vera equalia, et alternis equalia sunt. Et g a ergo recte g b est equalis. Tres ergo recte g a, a b, b g equales sibi invicem sunt. Equilaterum ergo est a b g trigonum. Et constitutum est super datam rectam terminatam a b. Quod oportebat facere.[64]

As I claimed above, the four translations differs a lot from one another, both concerning the choice of words, phrases and constructions, as well as in terms of disposition and arrangment.

When it comes to orthography it is clear that the texts were written in a time before the classic ideal once more became the ideal one desired to immitate. Busard writes in the introductions to his publications of the translations made by Gerard, Adelard and the anonymous translator that he has made no attempt to keep the orthography found in the manuscripts[65]. He further writes that he has applied ti in front of vowels, but that ci is more common,[66] but that he has kept the orthography found in the manuscripts of Hermann’s translation[67]. Since all the translations were made in the twelfth century one could expect a similar language and orthography, and this is also the case. One of the many things they have in common is the use of e instead of the diphthongs ae and oe. It also seems that ci is the preffered spelling in front of vowels, rather then the classical ti. Adelard, on his hand, seems to write punto instead of puncto from time to time. In addition it is worth noticing that Adelard also confuses the gender of triangulus, by first writing triangulam, but later on both triangulum and triangulus (the former being feminine, the two latter masculine). Similar to Adelard Hermann also writes spacio (ci instead of the classical ti), and e instead of ae, but apart from that he mainly follows classical orthography.

A last interesting feature about Gerard’s text is the usage of manuscripts. Bussard claims that Gerard“based his labors on an Ishaq-Thabit text that already contained material drawn from al-Hajjaj I.[68]”I, on my hand think he also had access to at least one Greek manuscript, as some of his propositions contains parts that are nowhere to be found in Arabic manuscripts, but only in the Greek original text. A good example occurs in proposition II.2[69]: In this proposition we see the following translation made by Adelard: “Si fuerit linea in partes divisa, erit illud quod ex ductu ipsius in omnes partes suas sicut quod ex ductu ipsius in seipsam.[70]” Hermann applies: “Omnis linea in partes divisa, quod in semetipsam ducta generat, equum reddit eis, que ex ductu eius in omnes partes suas procreantur.[71]” Whilst Gerard writes: “Si linea recta fuerit divisa in divisiones, quocumque modo accidat, superficies rectorum angulorum que comprehenduntur ab illa linea tota et ab unaquaque ex divisionibus eius quadrato ex tota linea facto sunt equales.[72]”A sentence that matches the Greek: “᾿Εὰνεὐθεῖαγραμμὴτμηθῇ, ὡςἔτυχεν, τὸὑπὸτῆςὅληςκαὶἑκατέρουτῶντμημάτωνπεριεχόμενονὀρθογώνιονἴσονἐστὶτῷἀπὸτῆςὅληςτετραγώνῳ.[73]” In this example we can clearly see that Gerards translation is closer to the Greek original than the other two translations: Especially because he applies parts that are only to be found in the Greek original, for instance quocumque modo accidat: A sentence that goes together with ὡςἔτυχεν, but is lacking in the other translations. In addition he writes  que comprehendunturwhere the Greek text says περιεχόμενον. This is yet another word only occuring in the Greek original, and I think that this shows that it is most likely that Gerard has had a Greek manuscript avaiableduring the translation process.

Conclusion

Despite being all translated in the twelfth century, all of the translations discussed and analyzed in this paper differs from one another to a certain extent. They all differs a lot when it comes to the construction of oblique clauses, and in this matter it seems that they have chosen each their preferred solution. There also seems to be a bit confusion concerning the gender of nouns, especially in the first proposition (among other places) in Adelard’s translation. Concerning Gerard’s translation it is likely that he had a Greek manuscript in addition to the Arabic ones. It was showed that they all choose each their way of arranging each proposition, even though one is able to find certain similarities between Adelard and Gerard, and to a certain extent also between Hermann and the anonymous translator. In short: They all differ from one another in every aspect, even though there are many similarities as well; but this is, after all, what you have to expect from different translations, despite the fact that they are all made within the same century.

The Arabic Text of the Elements:

There is still no published edition of the Arabic translation(s) of Euclid's Elements. Some passages have been edited as part of doctoral theses and in scholarly articles, and a few facsimilies and 19th-Century editions of al-Ṭūsī's Taḥrīr ("Edition/Recension") have been published. Most of this is not easily available, and to tackle the text itself we must work from the manuscripts themselves. This online edition is based on the Uppsala Manuscript (Uppsala Universitetsbibliotek, MS. O. Vet. 20), but information gathered from sporadic use of other MSS will also be added, and mentioned in the notes when not trivial (e.g., minor lacunae checked against other MSS will not necessarily be indicated). The idea is to provide a "diplomatic edition" of the Uppsala MS. The online text will be constantly revised and annotated, and as the project evolves we hope to make as much as possible of the Arabic material on Euclid's Elements available on the net.

The Greek text was apparently first translated by al-Ḥaǧǧāǧ ibn Yūsūf ibn Maṭar, who was active during the reign of Hārūn al-Rašīd (AD 786-809). This text was later referred to as the Hārūnī edition. al-Ḥaǧǧāǧ later translated the text again, during the reign of al-Ma’mūn, and this version was accordingly named the Ma’mūnī edition. A new translation was produced by the more familiar translator Isḥāq ibn Ḥunayn (d. AD 910), whose translation was later revised by Thābit ibn Qurra (d. 288/901), who seems to have had access to other Greek manuscripts for his revision (De Young 1984: 149). The "standard" version, so to speak, then became the Isḥāq-Thābit version, which was the basis of Naṣīr al-Dīn al-Ṭūsī's (d. 672/1274) recension/edition (Taḥrīr).

Notes on the Arabic translation:

• Editorial additions and comments (occasionally chapter headings) are marked by square brackets: [الحدود]
• Textual reconstruction (of lacunae etc) is marked by angled brackets: <الأشياء>
• The letters referring to the figures (diagrams) are marked in the text by parentheses ( ), following the convention of modern Arabic editions (the MSS usually have a line above these letters).
• The letters are also written separately (e.g. ب ج and not بج etc.), although in the Uppsala MS they are usually written as a group.
• Other modern editorial interference like commas, stops and dashes is avoided.
• Pending a unified system for page and line references in the Bibliotheca Polyglotta or at least in the BPG, the Arabic texts (beginning with Euclid's Elements) will be given in their input form in a purely ad-hoc but consistent notation: every line in the MSS/editions, at least when lines are part of the input data, is marked with data for page and line within curly braces enclosed by square brackets, thus: [{p,l}]. This "layout" will surely change in the future...
• Ṭūsī's own comments to the Euclidean text is marked by double curly braces: {{ الطوسي }}.

Abstract and concrete models of Euclidean geometry in light of proposition II 1 of the Elements: Arabic versions of Tūsī and Nayrīzī.

Serena Baldari

Introduction

The Elements is a scientific treatise, an organization of geometrical and mathematical principles, proven in tight lines of logical arguments. But it is also a textbook: a course for learning and teaching geometry, not only as a purely theoretical discipline, but as a practical and applicable art. The Elements has influenced the development of scientific methods and the way we think about mathematical knowledge for over 2000 years, and at the same time it has widely been seen as a part of the basic curriculum of education. It is this dynamic, that makes the Elements so fascinating, that also makes it problematic for philosophers. For what is the nature of Euclidean geometry? Is it empirical or practical, general or particular? And what are the essential properties of the Euclidean geometrical models? Are they abstract or concrete? Do they simply exist, or are they constructed?

Understanding the connection between the form and the content of the propositions is necessary in order to fully appreciate Euclid’s work. For if one studies the Elements only to grasp the geometrical concepts proposed, the style of writing will seem repetitive and superfluous; and someone only interested in the Elements as a formal system, will find it to be dubious and unsatisfactory. The satisfied student of Euclid is he who looks for “a form of unity beneath the diversity of experience” (Berlinski 2013:11).

In Part 1 of this project I present some of the questions that arise in trying to study book II of the Elements. I look specifically at the first structural elements of a Euclidean proposition, the protasis and ekthesis-diorismos. These represent two ways of portraying the proposition: in the first it is stated generally, in the second it is described as an example. What is the difference between the two, is there any at all, and if not, why are they both necessary?

Another aspect of studying Euclidean geometry is language. The Elements is one of the most translated, edited and commented works of all time, and the fact that we do not have Euclid’s original manuscripts adds to the interest of deciphering the various available versions. In part 2, in the light of the questions and topics discussed in part 1, I make some observations about the protasis and ekthesis-diorismos of the Arabic versions of two Persian medieval scholars: An-Nayrīzī (9^th-10^th century) and at-Tūsī (13^th century). The analysis will be on proposition 1 in book II, and not representative of the whole Elements.

The Elements contains 13 books and more than 460 propositions, all based on 5 common notions (κοιναί έννοιαι) and 5 postulates, or axioms (αιτήματα). The books of the elements do not only deal with geometry, but also arithmetic and theories of proportions and magnitudes, but the subject of books I and II is plane rectilinear geometry (Mueller 1981:viii).

PART 1
The structure of Euclidean propositions

Euclidean propositions can be divided into two categories: problems (problemata) and theorems (theoremata). Problems are tasks, instructions for something to be done; while theorems are claims, assertions that something is true. This distinction is not found in the Elements itself, but is in use among the Greeks and discussed in depth by Proclus. “Every problem” he says, “has some theory in it; but the reverse is not true, for demonstrations in general are the product of theory.” (Proclus 1970:65) This again illustrates the fascinating and problematic dynamic of the Elements as textbook and treatise. However, all the propositions of the Elements follow the same general structure, and their parts are clearly distinguishable.

The protasis, or enunciation in English, is the general statement. It states, as generally and concisely as possible without the use of symbols, the matter that is to be proven in the proposition. In book II usually the protasis is formulated as a conditional sentence in Greek, starting with ἐὰν “if…”. This is reflected in most Latin translations with si… although not in the Arabic. An-Nayrīzī maintains a consistent pattern, starting with كل followed by a singular indefinite: “each, every…”, although a conditional sentence introduced by اذا also occurs.

After the protasis comes the ekthesis (exposition). This is the example, where the protasis is given a face, a particular identity, and its elements are named. The example itself, though, is meant to be general, and any given name representing a point, line or other structure is meant to be inter-changeable with any other name, symbol or size. In the Arabic versions, as also in some of the Latin, this part is usually explicitly introduced as an example: مثلا (mathalan) “for example” in at-Tūṣī, and مثاله ان (mithāluhu anna) “it’s example/likeness (is) that…” in An-Nairīzī.

This idea that the example is general is strengthened by the diorismos (specification). Here the task to be carried out (for problems) or the truth to be proved (for theorems) is stated on basis of the ekthesis. In theorems the diorismos will start with λέγω, ὅτι “I say that…”, which is reflected in An-Nayrīzī’s Arabic: فاقول ان, but not in at-Tūṣī’, who often skips the diorismos altogether.

Then we have the kataskeue (construction), and apodeixis (proof), and finally the proposition is summarized in the sumperasma (conclusion). In book II, only propositions 11 and 14 are problemata, ending with ὅπερ ἔδει ποιῆσαι “which had to be done” (Q.E.F. from the Latin quod erat faciendum). All other are Theoremata and end with ὅπερ ἔδει δεῖξαι “which had to be proved/demonstrated” (Q.E.D, quod erat demonstrandum). An-Nayrīzī has an equivalent ending وذلك ما اردنا ان نبين (wa-dhālika mā aradnā an nubayyina) “and this is what we wanted to demonstrate”. The verb بيّن, form II of بان has the meanings “to make clear, evident; to show, demonstrate, elucidate”, and is the equivalent of the Greek δείκνυμι “to show, point out, reveal”. At-Tūṣī has a similar construction in the conclusion, but without using بيّن. In addition, this Q.E.D equivalent comes at the start of the sumperasma, announcing it, instead of at the end. In general, the structural elements of At-Tūṣī’s version are more irregular, less defined and often even omitted. Although An-Nairīzī has a much more consistent structure, it also has exceptions.

Logical problems of the Elements

Mueller argues that the ekthesis, being the beginning of the proof, makes a justification of the generality of the proof necessary. The generality of what is being proven is simply stated as a fact in the protasis, repeated in the sumperasma, and insisted on in the diorismos. But the proof itself, with basis in the ekthesis, is really a particular example, and there is made no conscious logical effort to justify the link between the particularity of this example and the generality of the proposed claim. The generality is assumed instead of proven. (Mueller 1981:12-14)

What seems to be lacking in Euclid’s proof, seen with modern eyes, is a method of verifying that the example given in any proposition actually corresponds to a universal truth. In Mueller’s argument it all boils down to the ancient Greeks’ perception of geometrical objects. Geometry is for them not seen as an ordered system of space, in which the objects are related to each other by power of their absolute existence. For Euclid every relationship between two objects is created by constructing them. We draw a line between the points A and B. The line AB is constructed onto the two points, and is therefore dependent on them. Berlinski expresses this characteristic in the preface of his book as an “intimate dependency among parts” (Berlinski 2013:xii).

A recurrent problem for philosophers is that Euclid seems not to have been conscious about what kind of geometrical system his models belonged to. They are constructed and they are general. Just like that, very straightforward. There is no explanation to whether the models are by nature isolated and then connected by construction, or are in relation to each other by virtue of absolute existence in a common system. Why doesn’t Euclid bother to make this distinction?

Berlinski proposes that Euclid’s conclusions are not really syllogistic, but that the logic rather lies in the self-evident connection between the statement made by the proposition and the particles used to express it: if so and so, then this and this, and also this, but not this, and therefore this etc. If this is true, then what does it mean for the translations of the Elements to other languages? There is no reason to assume that this crucial connection will automatically reappear in translated versions. Also it is not probable that the translators were aware or cared about the importance of such a link. And if they were, what difficulties could they have met in the process of recreating the same connection in the language of translation, what particles or grammatical constructions in Arabic could serve the same logical purpose as the particles used by the Greeks?

The Elements illuminates a tradition where the power of the mind, mathematical intuition and common intelligence are taken for granted. This reminds me of Socrates’ dialogue with Meno. Socrates shows that even Meno’s slave boy who has not been taught any geometry has true beliefs about geometrical concepts. This is to prove that learning only makes sense when it is seen as remembrance of something already known to the soul. With this in the background, the straight-forwardness of Euclidean logic makes more sense.

For as all nature is akin, and the soul has learned all things; there is no difficulty in her (the soul) eliciting or as men say learning, out of a single recollection -all the rest, if a man is strenuous and does not faint; for all enquiry and all learning is but recollection. (Meno by Plato)

The nature of Euclidean geometry: abstract and concrete models

What marks the difference between an abstract and a concrete object? The first thing that comes to mind is sensory perception. The concrete is what we can see and touch. Like a book. If I think about a book, or close my eyes and envision it, obviously I can’t touch it. The book as a concept, however, is still concrete – only the thought or vision of it is not. But what if I envision something that is like a book, but is not like any existing material book that I can see and touch? For example I can think of a book with infinitely many pages. It can’t be found anywhere on earth, but it is still a book. Is it then concrete or abstract?

The Oxford Student’s Dictionary defines abstract art as art which does not represent an object, scene etc. in its true, normal or usual form. Since art can be seen as a form of visualization, this definition could apply to the infinite book example, thus making it abstract. In this sense the abstract is a distortion of reality and of truth. But then why does it seem so natural to still call it a book? I can say to anyone “think about a book with infinitely many pages” and assume him to understand what I mean by it, even though we both know that there is no such book. There is also another problem: how can we say that an infinite book is less true than a finite one? Robert Kaplan says that mathematics is “the art of the infinite” (Kaplan 2003). If the infinite is untrue, then where does that leave mathematics?

The same dilemma presents itself when analyzing the geometrical models of the Elements. A point, a straight line, a right angle; they are simple concepts that can be easily defined and understood. But among the shapes we can see and touch in the world around us there is nothing like them. Even so, Euclid requires them to be drawn, constructed and described. Berlinski draws attention to the same problem concerning equality. The fourth common notion of the Elements defines equal objects as coincident with one another. Concrete objects, he argues, cannot coincide perfectly, so they are never completely equal. And abstract models cannot be coincident, because placing one upon the other would require a movement of them in time and place, whereas they are by definition excluded from physical space. (Berlinski 2013:28)

In Berlinski’s analysis the Euclidean models seem to be excluded from both realms, concrete and abstract. But the models exist - whether their existence is absolute and objective, or dependently constructed – they must be something. One possible solution is described by Bertrand Russell as “transference of attention” (Berlinski 2013:28). The motion in space apparently required by Euclidean shapes can be perceived as a thought motion, instead of an actual one. The mind can visualize a movement of shapes in space, just like it can visualize a flying horse without attributing to horses the ability to fly. We can move abstract lines and triangles around by imagination, without having to attribute the ability of physical movement to abstract objects.

Another solution can be found in intuitionism (or constructivism), a direction within modern philosophy of mathematics in which existence and construction are not only compatible, but even inseparable. Absolute existence and independence of mathematical objects is seen as impossible, and the foundation of mathematics is the mental ability to understand a concept as constructed from another concept. But to call Euclid a constructivist would be to attribute to him a modern way of thinking. It is more likely, according to Mueller, that Euclid’s understanding of geometric objects is linked to the perception of geometry as a practical and applicative discipline. (Mueller 1981:15)

Also, pure intuitionism does not accept the concept of infinity, and Euclid obviously does, since he attributes the quality of infinity to his parallel lines in definition 23 of book I: …being produced indefinitely in both directions, (the straight lines) do not meet. The word translated here as “indefinite” is ἄπειρον. Kaplan claims that apeiron literally means “without boundary”, and is therefore also appropriately translated into “infinity”. Represented by this double meaning, the ambiguity of endlessness allows us to unite abstract ideas and concrete objects in our mind:

Just as we picture continuity in the material world by rocks between boulders, stones between rocks, pebbles between stones, and sand to fill in the crevices, so we see fractions in the spaces between integers – and for fractions “even smaller” means denominators becoming infinitely large. (Kaplan 2003:75)

Several philosophers have proposed the idea that the elementary particles, which all matter is made up of, can be seen as the physical equivalents of geometrical points. This conceptual bridge between abstract and concrete allows us to think differently about the geometrical models of the Elements.

According to the dictionary, something abstract is “thought of separately from particular examples”. It is “the idea of general nature of something, rather than its real or outer form” and “regarded in a general or theoretical way”. Going back to the problem of generalization of Euclidean propositions from the seemingly particular example of the ekthesis-diorismos, we can now see that the claim of generality insisted upon by the propositional structure can be confirmed by the abstract nature of the models. On the other hand, the lack of an explicit generalization method allows the models to also have a concrete nature. Just like the abstract of an essay presents it and summarizes it, without undermining it; the protasis carries the abstract geometric shapes through the concrete lines of the geometer’s straightedge and into universality.

Understanding the Elements

As already mentioned, mathematical intelligence is assumed by the Greeks. But how much intelligence is required of us in order to study the Elements? As all the axioms and definitions (and even common notions) needed to follow the arguments are included, no prior knowledge of geometry is required. Berlinski compares studying an Euclidean proposition to climbing a mountain. Making each step is easy, but making all of them is not. Also, and more interestingly, he claims that steps cannot be skipped (Berlinski 2013:59).

One question is whether we are able to understand the propositions without the illustrations that come with them. Probably for the common man it is not. Even recreating the illustrations based only on the written text is extremely difficult. Another question is to what degree it is possible to grasp what the propositions are proving from only reading the protasis. Mueller writes that it is the absence of variables that creates in the protaseis “sentences so complicated it is difficult to believe that anyone could understand them without reading the ekthesis and diorismos” (Mueller 1981:12).

Joseph Mazur, also with reference to Plato, points out that our understanding of numbers is not dependent upon an axiomatic arithmetic system. The capacity for understanding numbers and how to use them is prior to our need to systemize them. We make abstract systems for numbers, he says, but the consequences following from them cannot be disconnected from the reality we see around us. And that in the end it is God who has created us with the ability to understand what He wants to show us. (Mazur 2005:90-91)

PART 2

The Arabic of the Elements

The classical period of the Arabic language started in the late 8^th century and ended before the end of the 10^th, the Arabic philology being developed during the 9^th century (Fischer 2002). So about a period of 200 years between 800 and 1000 A.D.. An-Nayrīzī appears right in the middle of this formative period, while at-Tūsī is later and must be considered post-classical.

The formal Arabic language is mostly preserved and the differences between classical Arabic and modern written Arabic are few. The morphology, syntax and word meanings remain the same. For this reason it is not a problem to use a reference grammar of classical Arabic combined with a dictionary of modern written Arabic in the analysis of these texts. But through time new words and expressions have appeared, and new constructions have been taken in use, like in the development of any language. Therefore the historical context in which the texts are written cannot be ignored.

Keeping this in mind, it is still possible to study them and say something about them, in humility and with some reservations, without much knowledge of the worlds in which they were produced.

Book II

A characteristic of book II of the Elements is that the propositions easily translate into algebraic expressions, and are considered by some to be geometric representations of algebra. In modern mathematics algebra is seen as primary by virtue of simplicity and usefulness. (Mueller 1981:42)

Book II is the smallest book of the Elements in number of propositions, with only 14 of them. It deals with many of the same elements as book I, under the common label of plane rectilinear geometry, only introducing the definition of a parallelogram.

Proposition II.1 concerns the quadrilateral figures made up of two lines and their parts. Its algebraic expression is a(b + c +...+ n) = ab + ac +...+ an, and its protasis is formulated as following:

Ἐὰν ὦσι δύο εὐθεῖαι, τμηθῇ δὲ ἡ ἑτέρα αὐτῶν εἰς ὁσαδηποτοῦν τμήματα, τὸ περιεχόμενον ὀρθογώνιον ὑπὸ τῶν δύο εὐθειῶν ἴσον ἐστὶ τοῖς ὑπό τε τῆς ἀτμήτου καὶ ἑκάστου τῶν τμημάτων περιεχομένοις ὀρθογωνίοις.

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

Being short and simple, proposition 1 is not a good choice to represent the geometrical complexity of Euclidean propositions, but it suits for shedding some light on the philosophical problems discussed in part 1 and illustrating and comparing the styles of writing of the two Arabic versions.

Proposition 1 of book II

At-Tūṣī
Manuscript: Istanbul

The protasis:

سطح الخط فى خط آخر يساوى جميع سطوحه فى اقسام ذلك الخط *

سطح (saṭḥ) “surface” comes from the verb saṭaḥa “to spread out, unfold, flatten”. In the etymology of the word lies also the adjectival meaning of “external, exterior, superficial” (سطحي). In Euclid’s definitions a surface is “that which has length and breadth only”, “the extremities of a surface are lines” and “a plane surface is a surface which lies evenly with the straight lines on itself” (Definitions 5-7, book I).

خط (khaṭṭ) “line, stroke” from the verb khaṭṭa “to draw, trace a line”, “design” and also “prescribe”. The meanings of the root include something that is particularly sketched or written by hand; and a plan, layout, or guiding principle. At-Tūṣī seems to find no need to specify that we are talking about a straight line (خط مستقيم).

فى (fī) “in, at, on, by, with” should most likely be translated as “on”, with the implied meaning of “perpendicular to”. Assuming, maybe, that in this is the most obvious way to place a plane surface upon a straight line. At least this makes sense if we already know that our purpose is to construct a right angled figure.

آخر (‘ākhar) “another, one more” belongs to the root for delay, “to be late, slow” or “fall behind”. So whatever comes after, be it “the other” (al-‘ākhar) or “the end” (al-‘ākhir) is something that comes relatively late, it has “fallen behind”. This same connection can be found in English in “the hereafter” (al-‘ākhira), “hereafter” meaning “from now on, after this”.

سطح الخط is a definite iḍāfa, made definite as a construct by the article on the genitive, or possessor. Its translation is therefore “the surface of the line”, although “the surface of a line” seems more appropriate, as no line has yet been defined. How do we understand this surface? We know by the root سطح that it is something “spread out” or “unfolded”. So we could also think of it as “the spreading out or unfolding of the line”: If the given one-dimensional line is spread out unto two-dimensional space, it will cover a surface. But this surface could go on forever and needs a limit, which is another line. The text says nothing about whether this line is equal or different from the first. It could simply be a matter of conciseness. The lines may be equal, but they don’t have to be. The proposition is true either way, so there is no need to spend time specifying the unspecified. But we can also interpret this line as different from the first, simply because it is not defined to be equal. In this case the text seems to assume that otherness stands for inequality. This supports the conception of the lines as parts of a spatial system; in their absolute existence, there is only one of each of them. Any equal object is the object itself. And any other object is different from it. This is characteristic, as we have seen, of abstract objects.

يساوى (yusāwī) “(it) equals, is worth” is a form III verb of sawiya, here in the 3^rd person singular masculine, imperfect indicative. Its subject is everything that precedes it in the sentence, essentially a surface and specifically the surface surrounded by two different lines, i.e. a rectangle. This rectangle is the equivalent of (and therefore has the same worth, value as)…

جميع (jamīʿ) “total, entire, all”, with the root signifying “to gather, join, combine”, should be followed by a genitive: “the total/joining of…”

سطوحه (saṭūḥihi) is the plural of سطح with an attached 3^rd person singular pronominal suffix. In construct with جميع this makes “the total of its surfaces”. But who’s surfaces? We established the subject to be “the surface of the line” and not “the line” itself. From the context it is obvious that the pronoun here refers to the line, or one of the two lines.

اقسام (aqsām) is the plural of qism, “part, division, section”. Qasama also means “to distribute, arrange, assign” in addition to “divide, part, split”.

So جميع سطوحه فى اقسام ذلك الخط is “the total of its surfaces on the cuts of that line”. The confusing part here is to know which line is which: The total of which line’s surfaces on the cuts of which line? Since the first line mentioned is definite, and the other is not, it would be reasonable to think that the pronominal suffix goes back to the first, and “that line” refers to the second. None of the two lines have been quantified, so they are interchangeable with one another, but confusing them may still be a problem for the argument of the proposition. This is what Mueller referred to, as mentioned, that the absence of variables in the protasis makes it difficult to fully understand.

Conclusive translation: The surface of the line on another line equals the total of its surfaces on the sections of that line.

The ekthesis-diorismos:

مثلا سطح ا فى ب ج يساوى مجموع ا فى خطوط ب د د ه ه ج التى هى اقسام ب ج

مثلا (mathalan) is the accusative of mathal “likeness, metaphor, example, model” used as an adverb “for example”. According to Wehr’s dictionary, mathalan is the equivalent of “e.g.” (exempli gratia) used in this case in the Latin version of Adelard.

مجموع (majmūʿ) “sum, total, collected” is the passive participle of jamaʿa. It has practically the same meaning as jamīʿ in the protasis.

خطوط (khuṭūṭ) is the plural of khaṭṭ.

التى (ʾallātī) is the feminine singular relative pronoun. It refers to khuṭūṭ because all non-human plural nouns are treated grammatically as feminine singulars. Here it should be translated, therefore, as “they that”.

The translation becoming: For example: The surface of A on BG equals the total of A on the lines of BD DE EG, those that (are) the parts of BG.
(I use the Latin equivalent of the letters in the translation for simplicity).

In سطح (ا) فى (ب ج) the lines are finally named. The first is called simply A (ʾalif) and the second BG (bāʾ- jīm). Two completely different notations for naming lines are used. One line is represented by a single letter, while the other is represented by two. In the case of BG the letters B and G represent the two points that are the extremities of the line; the points between which the line is found. The letter A, on the contrary, is not a point on the line; it is the name of the line itself as a whole. The difference in the naming of the lines is reflected in their different roles in the proposition. Line BG is going to be divided, and to distinguish its sections it is useful to use the two extremities as points of reference. Line A is not going to be cut, and there is no reason therefore to waste two letters on it, when one is enough. Concision is valued over consistency.

Since nothing about a line is mentioned here, the letter A could easily refer to the surface, translating سطح ا as “surface A” instead of “the surface of (the line) A”. But because we read the protasis we know that A and BG are lines. The same problem arises in مجموع ا. “The total of A” is really “the total of the surface of the line A”. But it is also possible that A refers to the surface and not to the line. That would solve both problems and also explain the difference in notation between A and BG. Since in the Greek A is the line, and this manuscript of at-Tūsī’s Elements comes with no illustrations, I chose to eliminate this possibility for now. If we are still thinking about A as the line, we can attach the new labels given in the ekthesis-diorismos to the unnamed objects of the protasis. It gives us:

سطح الخط (ا) فى خط آخر (ب ج) يساوى جميع سطوح (ا) فى اقسام ذلك الخط (ب ج)

When reading the proposition we can do this fusion in our minds and create a mental image of the figure, even without illustrations. This allows for the abstract statement and the concrete definition to maintain their separate individual identities, and simultaneously produce together an understandable unified concept.

An-Nayrīzī
Manuscript: Codex Leidensis 339

The protasis:

كل خطين مستقيمين يقسم احدهما باقسام كم كانت فان السطح الذى يحيط به الخطان مساو لجماعة السطوح التى يحيط بها الخط الذى لم يقسم وكل واحد من اقسام الخط الاخر المقسوم

كل (kull) followed by an indefinite noun: “every”. Interestingly, كل واحد (kull wāhidin) has the meaning of “every (single) one, each one”

خطين مستقيمين (khaṭayni mustaqimayni) “two straight lines” in the dual, genitive after كل.

يقسم (yuqsimu) imperfect indicative passive of qasama: “is cut, divided”.

احدهما (aḥadhumā) “one of the two”, subject of يقسم.

باقسام (bi-iqsāmi) “into segments”, stands adverbially to يقسم.

كم (kamm) “amount, quantity”; or the interrogative (kam) “how much” used as a relative pronoun.

كانت (kānat) “was, were”, f.sg. perfect.

كم كانت (kamm kānat) must correspond to εἰς ὁσαδηποτοῦν “in any number”. Another possibility is that it is an equivalent of ὡς ἔτυχεν, which in the Greek text is used in the ekthesis of the same proposition, but the meaning of kam/kamm expresses number, quantity, and not randomness.

فان (fa-ʾinna) “and verily, truly”, introduces a man clause.

الذى (ʾalladhī) m.sg. relative pronoun, referring to السطح.

يحيط (yaḥīṭ), imperfect indicative of حاط “to surround, encircle, enclose” followed by ب. The verb also means “to guard, protect, have custody of”.

At first sight is seems more appropriate to think of this as yuḥīṭ, the passive form, its subject being الذى referring to السطح. But then it does not make sense to have it preceding به. Bi introduces what is being encompassed, and the masculine singular suffix can, as far as I can see, only refer to السطح/ الذى. The subject of يحيط should therefore be the following الخطان, which does then not have to agree with the verb in number, as is common in verbal sentences (where the verb precedes the subject).

This makes sense if الذى يحيط به الخطان is subordinate to the main clause مساو السطح “the surface (is) equal (to…)”.

مساو (musāwin) “equal, equivalent”.

لجماعة (li-jamāʿtin), جماعة is “group”. Why this word is used instead of جميع as in at-Tūsī is not clear. Perhaps there was a greater inclination towards metaphorical interpretations in the classical period, where group is understood as equivalent to accumulation, totality.

جماعة السطوح : “the group of surfaces”.

The clause التى يحيط بها الخط follows the same argument as earlier; it specifies which surfaces: “the ones that the line encloses them”. And from that we can ask: which line? The next relative clause answers the question: الذى لم يقسم.

لم يقسم (lam yaqsam) is the negated jussive: “was not cut, divided”.

كل واحد (kull wāhidin) we already mentioned: “every, each one”, with the meaning here of “each and every one”.

من (min) is here best translated as “of”.

اقسام الخط : The segments of the line. Now which line? الاخر “the other”, namely:

المقسوم (al-maqsūm), “the dividend”, which could also be translated here as “the segmented”.

Conclusive translation: Every two lines (where) one of them is cut into segments, how (ever) many they be: Verily the surface, the one which the two lines enclose it, (is) equal to the group of surfaces, they that the line encloses them, the one (line) which was not cut, and each and every one of the segments of the other line, (namely,) the segmented.

The ekthesis and diorismos:

مثاله ان خطى ا ب ج مفروضان وقد قسم خط ب ج على نقطتى د ه

فاقول ان السطح الذى يحيط به خطا ا ب ج مساوى لجماعة السطوح التى يحيط بها خط ا واقسام ب د د ه ه ج

مثاله (mithāluhu) “its example”.

ان (anna) “that”, conjunction followed by subject in the accusative case.

خطى (khaṭay) is the dual accusative in the construct state. خطى ا ب ج is an iḍāfa with A and BG as proper names.

مفروضان (mafrūḍāni) in the nom. dual: “supposed, assumed, premised”. This is the predicate of خطى ا ب ج.

قد قسم (qad qasama) “has been cut”. The particle qad with the perfect indicates an action that has or had already been completed, similar to the English pluperfect (past perfect).

على (ʿlā) “on, at”.

نقطتى (nuqṭatay) is the construct dual genitive of نقطة, here with the meaning of “point”.

Then comes the diorismos, clearly introduced by فاقول ان (faʾaqūl ʾanna) “and I say that…” And the following sentence is fundamentally equal to the protasis, only that the objects now are associated with the proper names assigned in the ekthesis. The only difference is that we find is in مساوى (musawiya) instead of مساو. I am not able to find a difference in meaning between the two, and there is no reason that there should be, but it is curious that an-Nayrīzī chooses to use different words when the sentences otherwise are so identical.

Conclusive translation:

Its example (is) that the two lines of A, BG are assumed. And the line of BG has been cut at the points of D, E.

And I say that the surface, the one which the two lines A, BG enclose it, (is) equal to the total of the surfaces, the ones that the line A encloses them and the segments of BD, DE, EG.

Final comments

In an-Nayrīzī’s text the combination of protasis and ekthesis is done in the diorismos, while in at-Tūsī’s version the ekthesis and the diorismos are fused completely together, and we had to do this combination ourselves, by attaching the names given in the ekthesis to the general description in the protasis. To say that he skipped the one or the other (ekthesis or diorismos) makes no sense, because it has the exempli gratia element of the ekthesis, but instead of just naming the objects, he states what their roles are in the proposition, which is the function of the diorismos.

In light of this small analysis an-Nayrīzī seems to be part of a tradition where translators followed more closely the Greek text, and we can even recognize corresponding expressions like كم كانت and εἰς ὁσαδηποτοῦν. At-Tūsī’s text seems to be an adaptation of the Elements for another purpose than simply producing a translation. Maybe because such direct translations already existed by that time. Even though at-Tūsī simplifies the text, he still maintains the division between protasis, general statement, and ekthesis, particular example. This is to strengthen the idea that both are actually necessary, they are two aspects of the same thing, but different.

Does this help us to answer any philosophical questions? - Probably not. We can’t explain how the spirit and body are the same and yet different, just like we can’t explain how Euclid’s models can be both abstract and concrete, equal and different at the same time. But to discover that the same problem presents itself in several disciplines is reassuring. It means that we are not looking at random coincidences. It means that there is “a unity beneath the diversity of experience”. It means there is hope of one day knowing the Truth.

References and literature list

All text of the Elements in Greek, English and Arabic is from Bibliotheca Polyglotta.

Arabic Dictionary: Hans Wehr Dictionary of Modern Written Arabic (Fourth Edition).

Greek Dictionary: The Online Liddell-Scott-Jones Greek-English Lexicon (http://www.tlg.uci.edu).

English Dictionary: Oxford Student’s Dictionary (Second Edition).

Fischer W. 2002: A Grammar of Classical Arabic. Third revised edition translated by Rodgers J.

Abu-Chacra F. 2007: Arabic: An Essential Grammar. Routledge London and New York.

Meno by Plato: Translated by Benjamin Jowett (http://classics.mit.edu/Plato/meno.html)
 

Berlinski D. 2013: The King of Infinite Space: Euclid and his Elements. Basic Books, New York.

Kaplan R. and E. 2003: The Art of the Infinite: Our Lost Language of Numbers. Penguin Books, London.

Mazur J. 2005: Euclid in the Rainforest: Discovering Universal Truth in Logic and Math. Pi Press, New York.

Mueller I. 1981: Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. MIT Press, London.

Proclus 1970 (Translated by Morrow, second edition 1992): Proclus. A commentary on the First Book of Euclid’s Elements. Princeton University Press, Oxford.

Euclidis Elementa, Bod: The 800 A. D. MS of the Elementa kept in the Bodleian Library, Oxford, of which there is a scan attached to the BP version, which has been made available from the Max Planck Insitute for the History of Science on the internet.

Heiberg, J.L. et Menge, H., Euclidis Opera Omnia, Leipzig: Teubner, 1883-88. This edition has been revised in J.L Heiberg (ed.), E.S. Stamatis (rev.), Elementa, 5 vols., Stuttgart, 1969-77.

Myungsunn Ryu: ΣTOIXEIA EΥKΛEIΔΟΥ, internet pdf edition, 2004.

MS in Bibliothèque Nationale, Fonds Latin 7373.

Edition in Busard, H. L. L. The Medieval Latin Translation of Euclid’s Elements: Made Directly from the Greek. Franz Steiner Verlag Wiesbaden GMBH, Stuttgart. 1987.

Murdoch, J. E. Euclides Graeco-Latinus, A Hitherto Unknown Medieval Latin Translation of the Elements Made Directly from the Greek. In: Harvard Studies in Classical Philology, 71. 1966. 249-302.

Heath, Thomas, tr., The Thirteen Books of Euclid's Elements. Dover Publications 1956, reprint of the second edition of 1925. The first edition is from 1908.

The Rekhâgaṇita or Geometry in Sanskrit composed by Samrâḍ Jagannâtha. Ed. Harilâl Harshâdarâi Dhruva, vol. I-II, Bombay Sanskrit Series no. lxi-lxii, Bombay 1901-2.

MS in Bibliothèque Nationale, Fonds Latin 16201.

Busard, H. L. L. The First Latin Translation of Euclid’s Elements Commonly Ascribed to Adelard of Bath. Universa, Wetteren. 1983.

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Busard, H. L. L. The Translationof the Elements of Euclid From the Arabic Into Latin by Hermann of Carinthia (?). E.J.Brill, Leiden. 1968.

CODEX LEIDENSIS 399,1: Euclidis Elementa ex interpretatione Al-Hadschdschadsch cum commentariis Al-Narizii. Arabice et Latine ediderunt R. O. Besthorn et J. L. Heiberg, Hauniae MDCCCXCIII-MCMXXXII.

Busard, H. L. L. The Latin Translation of the Arabic Version of Euclid's Elements Commonly Ascribed to Gerard of Cremona. E.J.Brill, Leiden. 1984.

Jihe yuanben 幾何原本; the Chinese translation of Euclid's Elements (Books I-VI) by Matteo Ricci 利瑪竇 and Xu Guangqi 徐光啟, Beijing 1607.

Euclides, Clavius, Christoph. Euclidis elementorum libri XV : Accessit XVI. De Solidorum Regularium comparatione. Omnes perspicuis demonstrationibus, accuratisque scholiis illustrati.  Romae, 1574.

Euclides, Clavius, Christoph. Euclidis elementorum libri XV : Accessit liber XVI. De Solidorum Regularium cuiuslibet intra quodlibet comparatione. Omnes perspicuis demonstrationibus, accuratisque scholiis illustrati : nunc quarto editi, ac multarum rerum accessione post primam editionem locuplatati.  Apud Aloysium Zannettum, 1603.

Jihe yuanben 幾何原本. The first 6 books of Euclid's Elements, translated to Chinese by Matteo Ricci 利瑪竇 [口譯] and Xu Guangqi 徐光启 [筆受]. Revised edition 明萬曆三十九年增訂本, 1611.

Engelfriet, Peter M. Euclid in China. The Genesis of the First Chinese Translation of Euclid’s Elements Books I-VI, Brill: Leiden 1998 (Sinica Leidensia, Volume: 40).

Input is done by Jens Braarvig (Greek, English, Arabic, Sanskrit, Latin, Chinese), Andrea Bréard (Chinese, Latin, Greek), Amund Bjorsnes (Arabic, Greek), Ciara Ebrahimpour (Persian), Eivind Lønaas (Latin), Karina Graj (Sanskrit), Serena Baldari (Arabic) and by others participating on the Euclid reading groups in The Department of Cultural Studies and Oriental Languages at Oslo University and in the Department of Classical World and Asian Cultures at the Friedrich-Alexander-Universität Erlangen-Nürnberg.