Marc Moyon
email: marc.moyon@unilim.fr

The Liber Restauracionis: a Newly Discovered Copy of a Mediaeval Algebra in Florence

Keywords: Algebra, Mediaeval Studies, Translation, Critical Edition, Manuscript, Algebraic Symbolism 01A30, 01A35, 01A75 Algebra, Mediaeval Studies, Translation, Critical Edition, Manuscript, Algebraic Symbolism 01A30, 01A35, 01A75

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Our paper concerns the appropriation of the sciences of Islamic countries by European scholars. It focuses on mediaeval algebra through a single text that I call the Liber restauracionis [Book of Restoration]. After our thorough study of the text in the Revue d'histoire des mathématiques [START_REF] Moyon | La restauration et la comparaison, ou l'art de résoudre des équations quadratiques dans l'Europe latine[END_REF], we propose here a critical edition based on three copies from the fourteenth century (at the latest). Indeed, when I was correcting the proofs of my

Introduction

Our paper concerns the appropriation of the sciences of Islamic countries by European scholars. It focuses on mediaeval algebra through a single text that I call the Liber restauracionis [Book of Restoration]. After our thorough study of the text in the Revue d'histoire des mathématiques [START_REF] Moyon | La restauration et la comparaison, ou l'art de résoudre des équations quadratiques dans l'Europe latine[END_REF], we propose here a critical edition based on three copies from the fourteenth century (at the latest). Indeed, when I was correcting the proofs of my aforesaid article, I received confirmation of the discovery of a new copy of the text in the Biblioteca Medicea Laurenziana of Florence.

First, I come back to the first two copies that we knew previously in order to present in detail the third one. I then summarize the main results of my study of the Liber restauracionis. The whole argument is published in [START_REF] Moyon | La restauration et la comparaison, ou l'art de résoudre des équations quadratiques dans l'Europe latine[END_REF]) with a French translation of the text. Finally, I give a critical edition taking into account the Florentine copy, with a more complete critical apparatus than that proposed in [START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF]).

The manuscripts

2. 1. Biblioteca Apostolica Vaticana,Vat. Lat. 4606 : (V) According to (Van Egmond 1980, 220-221), this paper codex (125 fol.) is from the middle or the second half of the fourteenth century. The Liber restauracionis (fol.72r-77r) is the third text of the codex including six mathematical texts. Five of them (including ours) are on computation and one of them deals with practical geometry. We should note that the algebraic parts of the Fibonacci's Liber abbaci is well-represented with copies of the fourteenth chapter and extracts from the fifteenth chapter.

Our text is entirely written in thirty-two to thirty-eight long lines, on each page, by a single scribe, in gothic hand. Geometrical diagrams (with letters) and tables are in the margins. Many marginal notes -references to Euclid's Elements -appear written by another hand, later than the principal scribe (in a cursive hand). Sections are clearly divided thanks to litterae notabiliores or minor initials (most of them are flourished initials in the margin). Titles of sections are added in marginalia. When the scribe does not write numbers in their literal form, he uses Arabic numerals in their classical mediaeval western forms [START_REF] Beaujouan | Étude paléographique sur la 'rotation' des chiffres et l'emploi des apices du x e au xii e siècle[END_REF][START_REF] Burnett | Indian Numerals in the Mediterranean Basin in the Twelfth Century, with Special Reference to the 'Eastern Forms[END_REF]. Particularly noteworthy is the writing of a fraction with an horizontal bar as in Maghrebian mathematics [START_REF] Djebbar | Le traitement des fractions dans la tradition mathématique arabe du Maghreb[END_REF][START_REF] Moyon | L'arithmétique des fractions dans l'oeuvre de Fibonacci: fondements & usages[END_REF].

This was the first copy of the Liber restauracionis known in the historiography. It was used, in the middle of the nineteenth century, by Baldassarre Boncompagni, who edited the text [START_REF] Boncompagni | Della vita e delle opere di Gherardo Cremonese traduttore del secolo duodecimo e di Gherardo da Sabionetta astronomo del secolo decimoterzo[END_REF], and after him by Franz Woepcke (Woepcke 1854, 356-357). It has a title, written in red at the top of fol.72, which announces: Here begins the book, which is entitled algebra and almucabala, following the Arabs, and Book of restoration according to us. And it was translated by Master Gerard of Cremona in Toledo from Arabic into Latin. 1This title is the reason why Boncompagni, whose "publications were mainly devoted to reconstructing the channels of the transmission of mathematical knowledge from the Arab world to Christian Europe and to its chronology" (Fiocca 2017, 2), edited the text in his paper dedicated to the life and works of Gerard of Cremona (Boncompagni 1851, 412-435):

A translation from Arabic to Latin, made in Toledo by Gerard of Cremona of an important algebraic treatise, was found in the Vatican n • 4606 from the recto of the fol.72 to the fol.76. This version being unpublished, it appears to me useful to reproduce it here, with the translation into algebraic language of some rules, and the operations that are indicated in it. 2But, today and since at least Björnbo's works (Björnbo 1905, 241), we have to consider it as a pseudepigraphon. 3Finally, the manuscript (V) is significant because it was probably the original source4 of a vernacular translation -named here (U) -made in Naples, at the end of the fourteenth or the beginning of the fifteenth century: the Biblioteca Apostolica Vaticana, Urb. Lat. 291 (fol.34r-41v). 5 Its early date (less than 50 years later than the original) suggests a strong interest in the Liber restauracionis in the early fifteenth century Italian. This version was studied and edited in [START_REF] Franci | Una traduzione in volgare dell'al-jabr di l-Khwarizmi (MS. Urb. lat. 291 Biblioteca apostolica Vaticana)[END_REF].

2.2. University of Oxford, Bodleian Library, Lyell 52: (L) This parchment codex of 75 folios was written in Italy, in four hands (De la Mare 1971, 143-146). It formerly belonged to the Benedictine Abbey of Admont (shelfmark 612). It is slightly older than the previous one, and seems to be from the early fourteenth century [START_REF] Kaunzner | Über eine frühe lateinische Bearbeitung der Algebra al-Khwârizmîs in MS Lyell 52 der Bodleian Library Oxford[END_REF]Kaunzner 1987, 132;Hissette 2003, 139-140). It contains several mathematical texts, above all treatises on algorismus, namely texts describing how to use Indian (commonly called Arabic) numerals in arithmetical calculations. In particular, in addition to an Algorismus de integris abbreviatus (fol.35r-41v), it includes two versions of al-Khwārizmī's Kitāb fī h . isāb al-Hind [Book on Indian reckoning]: the Liber pulueris (fol.1r-20r) and the Liber Ysagogarum (fol.21r-34v) (Allard 1992, xxxvii, xli). Several independent astronomical texts (tables/Almanac) complete this codex. Last but not the least, it is interesting to note that the major part of this codex is from Arabic astronomical or mathematical writings.

The Liber restauracionis, copied by a single gothic hand, occupies eight folios of twenty-eight lines ruled in hard point (one column), from fol.42r to 49v. This is the second copy that we know. It was used by Wolfgang Kaunzner in his critical edition [START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF]. There is no short or running titles, nor are there marginal notes. Numbers are written in their Arabic form as in (V). Except in tables of numbers (fol.44v-45r), fractions are represented literally. All geometric diagrams are executed in the margins (fol.43v, 44r, 46r) and they are labelled with letters. The tables of numbers are in the text. We note only three litterae notabiliores, including the pen flourished initial U of the incipit (fol.42r) and two plain initials (fol.44v, 46r). As a result, the different sections and problems are not well-separated. There are no marginal notes.

Finally, we want to point out that, after a rough analysis, 6 Wolfgang Kaunzner draws our attention to a significant feature of the manuscript (L) in the history of Latin algebraic texts issued from al-Khwārizmī's Algebra.

Thus, the algebra of L -probably early 14th century -could be the oldest of all listed. Even though the numbers are, for the most part, already ciphered in L and V, the way the fractions are written points to an early date of composition. 76 In Kaunzner's list, two French manuscripts (Paris, BnF 7377A and BnF 9335) are dated from the fourteenth century while Hughes dates them from the thirteenth century (Hughes 1986, 224-226). In addition, he does not seem to know the Florence manuscript, also from the thirteenth century, referenced by Hughes (Biblioteca Nazionale Conv. Soppr. J. V.18, fol.80r-86v) and already known by Björnbo (Björnbo 1911/12, 221). For the textual tradition of Latin versions, see (Hughes 1986, 221-228;Hughes 1989, 11-13).

7 Demnach könnte die Algebra aus L -wahrscheinlich frühes 14. Jahrhundert -die So, (L) is certainly not "the oldest of all listed", nevertheless we can consider it as one of the older ones. 8 Only three manuscripts of Gerard's translation, at least, are older: for Barnabas Hughes, they are from the thirteen century (Hughes 1986, 221-228). Florence,Conv. Soppr. 414: (F) This is the newly discovered copy from the thirteenth/fourteenth century:

Biblioteca Nazionale Centrale of

The years 1263 and 1304 are given as the 'present year' in notes added by another hand on f. 60r and 63v. (Juste 2017a) Even if the year 1263 is uncertain (see below), the year 1304, confirmed by two different glosses, seems to be relevant. David Juste mentions:

The date 1304 is congruent with the script and is also confirmed by another gloss in the same hand and on the same page, which speaks of 1185 years elapsed since Ptolemy's epoch of 118 (!) A.D. (118+1185=1303). (Juste 2017b) Thus (F) is certainly the oldest known manuscript for the Liber restauracionis, which is not contradicted by palaeographic arguments (in particular for numerals). It could even be contemporary with the older codices of Gerard's version.

Before belonging to the Biblioteca Nazionale Centrale of Florence, the codex was in the Basilica della Santissima Annunziata (Crociani et al. 1983, 117-118). The parchment codex, consisting of 63 folios, was apparently written by a single hand. It consists of a mathematical and astronomical miscellany. One of the most important texts is undoubtedly the Almagestum parvum (fol.1r-45r) 9 which is a "compendium of the six first books of the älteste von allen aufgeführten sein. Die Zahlen sind hier, in L und V, zwar großenteils schon verziffert, aber die Art, wie Brüche geschrieben werden, weist mit auf die frühe Entstehungszeit hin. (Kaunzner 1986, 84) I thank Andrea Bréard for confirming my understanding of this quotation.

8 It is also interesting to note that only one known Arabic copy of the al-Khwārizmī's Mukhtas . ar would be older than (L). It is from Medina, thirteenth century (Rashed 2007, 85;Rashed 2009, 85).

9 In this respect, the codex is recorded in the database Ptolemaeus Arabus et Latinus. Manuscripts, see (Juste 2017a).

Almagest [of Ptolemy] with additional material from al-Battānī and others" following Lorch (1992, 407).

Our text consists of the last four folios (fol.60va-63vb) and is presented in two pre-ruled columns of roughly equal size, between forty-one and forty-five lines. The scribe wrote in gothic textura, used Arabic numerals and to a lesser extent a literal form (especially to designate all fractions).10 No fraction is written using the horizontal bar (as in (V)).11 Geometrical diagrams (with letters) and other tables are in the margins. The beginning of a paragraph is marked with colored pilcrows or littera notabilior (all are simple plain initials).12 This copy does not have short or running titles, nor an incipit as in (V).

There are four marginalia in those folios, probably written by other hands (at least for three of them) at the beginning (fol.60va/60vb) and at the end (fol.63vb) of our text. All four are quite difficult to read for palaeographical reasons, or due to a poor quality ink. 13 First of all, in the header of the fol.60v, the sentence (that hardly makes sense): "Nota quod radix est numerus multiplicatus in se ipsum census est" clearly refers to the definitions of the objects of algebra: radix [root for the Arabic term jidhr ] and census [for the Arabic māl, often badly translated in English by square] (Oaks and Alkhateeb 2005, 401-402). The second one is longer and more difficult to interpret, in particular because I do not know if it is written by the main scribe or another hand. Even if a distinctive sign, , in the text refers explicitly to this gloss,14 it refers itself to another text or another gloss apparently written in another codex. Another sign, , is used that does not correspond to any other passage in (F). The third one consists in a few words on the left margin, in the fold of the codex ; it is impossible to read. At last, the fourth one is a longer marginal note at the end of fol.63v: 24 lines in the second column followed by 4 full lines. Nevertheless, this important marginal gloss, due to a later scribe, is an astronomical note without any relation to our text but rather to the previous Ptolemaic texts. 15 Note that, as we mentioned above, this is the gloss that mentions the year 1263 "as the present year" but it seems likely that it is a scholium copied from another manuscript. 162.4. An attempt to establish a stemma

The relationships among the manuscripts (L), (F) and (V) may be described using several factors that take into account only the text and not, in a first step, the geometrical diagrams. Indeed, we know that the diagrams are executed separately from the text and often by another hand. I think that we have enough philological arguments not to take the risk of making uncertain hypotheses about the diagrams.

First, as we saw, (V) has a principal title and sectional titles all of which are missing in both (F) and (L). But, as most of those titles are interlinear or marginal, this could be an isolated decision of the scribe of (V). Nevertheless, we have to notice some key features about the texts which I consider as the most striking differences between the manuscripts. In particular, we have identified several discriminating passages, some of which could be omitted because of homoeoteleuton. (F) does not show, at least, two passages found in (L) and (V) (notes 79, 173) but, a contrario, shows a passage not found in both (L) and (V) (note 557). Now, (L) lacks three lengthy phrases (more than four words) found in both (F) and (V) (notes 24, 240 et 473). Finally, (V) lacks, at least, four long passages found in (F) and (L) (notes 4.2, 318, 395 et 510). Hence, we can be certain that none is a copy of any of the others.

Secondly, we will argue that both (F) and (L) could have a common ancestor (α), which is not the exemplar (β) of (V). I must begin by stating that numerical data is very stable in the three manuscripts, except for a passage where (V) stands out. The problem in (F) and (L) is 5x 2 = 80 whereas it is 5x 2 = 40 in (V) (note 74). Then, more generally, I examined the case of readings common to two of the three manuscripts 17 when considering only the terminological changes (omission or modification of at least one word). I did not take into account the interchange of the same word or words in a sentence, 18 differences due to orthography, grammatical inflection or conjugation 19 or even differences of words that could easily be explained by errors in the reading of the medieval abbreviations (mine or scribe's). 20 I also naturally excluded from this study the three or four passages that present reading difficulties. As a result, we have the following Table 1. We can conclude that more often than not (F) and (L) agree in a reading against (V), regardless of which reading is correct. Now, if we observe geometric diagrams and tables of numbers (Table 2), this idea is not contradicted.

Pair of Manuscripts

(F) & (L) (F) & (V) (L) & (V) Number of Common Readings 53 30 25
figures observations in manuscripts Mss. differing from the others Fig. 2 (

F) =(L) =(V) (F), (L) and (V) Fig. 3 (F) =(L) =(V) (F), (L) and (V) Fig. 4 (F)=(L) =(V) (V) Fig. 5 (F) =(L)=(V) (F) Fig. 7 (F) =(L) =(V) (F), (L) and (V) Fig. 10 (F)=(L) =(V) (V) Fig. 12 (F) =(L)=(V) (F) Fig. 13 (F) =(L)=(V) (F) Fig. 14
absent from (F) and (L) (V) Fig. 15 absent from (F) and (L) (V) Relying on the foregoing points, I propose the following stemma (fig. 1).

18 For example,"omnis numerus multiplex est unitatis" versus "omnis numerus multiplex unitatis est". 19 In particular, the doubling of letters : hiis/his, digittus/digitus, equipolet/equippolet... I do not report here grammatical differences in person, number or tense.

20 For example, perfluat or profluat.

I include the manuscript (U) as an interesting witness.

Classic results in the historiography

The Liber restauracionis has been considered as one of the translations of al-Khwārizmī's Mukhtas . ar fī h . isāb al-jabr wa-l-maqābala [Compendious Book on Calculation by restoration and balancing], made directly from Arabic to Latin. We have already seen, concerning (V), that the Liber restauracionis cannot be counted among the numerous translations of Gerard of Cremona. Following [START_REF] Hughes | The Medieval Latin Translations of al-Khwārizmī's Algebra[END_REF][START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF][START_REF] Hughes | Robert of Chester's Latin Translation of al-Khwārizmī's al-jabr[END_REF][START_REF] Franci | Una traduzione in volgare dell'al-jabr di l-Khwarizmi (MS. Urb. lat. 291 Biblioteca apostolica Vaticana)[END_REF], the translation is attributed to Guglielmo of Lunis, a scholar attached to the Italy of botteghe d'abaco [schools of abacus]. He could also be identified with Guglielmo of Luna, a translator of Arabic philosophical works, in particular those of Ibn Rushd (1126-1198), who flourished in Toledo in the first half of the thirteenth century and in Sicily between 1258 and 1266 [START_REF] Hissette | Guillaume de Luna ou de Lunis ou Lunense: un même traducteur d'Averroès et de traités d'al-Jabr ?[END_REF][START_REF] Hissette | Des traductions doubles de Guillaume de Luna ou de Lunis[END_REF][START_REF] Hissette | d'al-Khwārizmī dans les mss Vat. lat. 4606 et Vat. Urb. lat. 291 et Guglielmo de Lunis[END_REF][START_REF] Delle Donne | Un'edita epistola sulla morte du Guglielmo de Luna, Maestro presso lo studium di Napoli, e le traduzione prodotte alla corte du Manfredi di Svevia[END_REF]. I will not recapitulate here the various arguments in favor of this attribution, I prefer to refer to my synthesis of scholarly works in [START_REF] Moyon | La restauration et la comparaison, ou l'art de résoudre des équations quadratiques dans l'Europe latine[END_REF].

In the recent literature, despite the cautious recommendations of (Hissette 1997, 128-129;[START_REF] Hissette | d'al-Khwārizmī dans les mss Vat. lat. 4606 et Vat. Urb. lat. 291 et Guglielmo de Lunis[END_REF], there seems to be a consensus to attribute the paternity of Liber restauracionis to Guglielmo of Lunis/Luna (and perhaps even that of the vernacular version). Here are some examples from the most recent works.

In his critical edition and French translation of the Mukhtas . ar, Roshdi Rashed specifies in a footnote that it exists in three Latin translations (Rashed 2007, 86;Rashed 2009, 86). In addition to those of Gerard of Cremona and Robert of Chester, he quotes that of "Guglielmo of Luna" for which, unlike the other two, he gives no bibliographic reference. 21In 2011, about the introduction of al-Khwārizmī's al-jabr into Europe, Nadia Ambrosetti states: "To mark the passage of this method to European culture, there are the three Latin translations made between the XIIth and XIIIth centuries in Spain, by Robert of Chester ( 1145), Gerard of Cremona (1170) and Guglielmo de Lunis (1250)(...)". 22Also, in their "History of Algebra from Antiquity to the early Twentieth Century", Victor Katz and Karen Parshall state that, in addition to Gerard and Robert, "there is at least one other independent translator into Latin of al-Khwārizmī's text, William de Lunis, who flourished in the first half of the thirteenth century" (Katz and Parshall 2014, 178).

In his recent synthesis on the introduction of Arabic Algebra in Europe, Ahmed Djebbar writes, with no other details than the reference to [START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF], that the Mukhtas . ar "has benefited from three Latin translations, by Gerard of Cremona (d. ca. 1187), Robert of Chester (ca. 1141) and Guglielmo of Lunis (13th century)" (Djebbar 2016, 104). Finally, it is also in these terms that Menso Folkerts and Barnabas Hughes consider these three names in their most recent Sourcebook in the Mathematics of Medieval Europe and North Africa (Katz et al. 2016, 103).

To my knowledge, only Jens Høyrup comes back to this attribution in a discrete note hitherto unnoticed. His conclusion is that, without new information, the Liber restauracionis must be considered as anonymous. Guglielmo of Lunis would be, for Høyrup, the translator of an Italian version now lost. I reproduce here the entire note:

This redaction is often supposed to be identical with a translation made by Guglielmo de Lunis. However, all references to this translation (except a false ascription of a manuscript of the Gherardo translation) borrow from it a list of Arabic terms with vernacular explanation which is absent from the present Latin treatise. It is a safe conclusion that Guglielmo translated into Italian; that his translation is lost; and that the present redaction is to be considered anonymous. (Høyrup 2010, 13)

My ideas

In this part, I summarize the main results of [START_REF] Moyon | La restauration et la comparaison, ou l'art de résoudre des équations quadratiques dans l'Europe latine[END_REF] where I analyse the text in detail.

Editing the Liber restauracionis, its filiation with the Mukhtas . ar is not in doubt. But can we consider this text as an Arabic-Latin translation of the al-Khwārizmī's text? Indeed, I do not believe that we can because of the many divergences between the two texts. My thesis is quite different. I notice, among others (the page references are those of my edition):

• the strong differences in the geometric proofs and diagrams for the algorithms (p.20-25) and the numerical identities with square roots: ( √

• the explicit reference to Euclid's Elements: p.20, 21, 23 and p.24 for geometrical proofs (Book II, see [START_REF] Moyon | La tradition algébrique arabe du traité d'al-Khwârizmî au Moyen Âge latin et la place de la géométrie[END_REF]Corry 2013, 655-659)), p.29 for the distributive property (again Book II, see (Corry 2016, 311-314)) and p.43 for reference to irrational numbers (Book X );

• the presence of new paragraphs (like the one in relation to the symbolism -see p.21) or paragraphs structured differently (as for the proportion -p.36);

• some numerical examples that are not necessarily those of the Mukhtas . ar (see, for example, Appendix A and Appendix B for the series of problems at the end of the text); and, more generally,

• the choice of a logical presentation that is more deductive than that of al-Khwārizmī himself. The author goes from the general rule to numerical examples, when these are given.

For me, it would therefore rather be a reworking, relatively more didactic than the Mukhtas . ar, which could be used to teach Algebra. Now, this reworking may have been written in Arabic, by a mathematician from the Islamic West, during the late Twelfth century or later (from the presence of algebraic symbolism and the fraction bar). In this case, the Liber restauracionis is de facto a Latin translation from an Arabic text (but not directly from al-Khwārizmī's Algebra) still unknown, and then Guglielmo of Lunis could be the translator.

However, the use of versification (p.17, 19, 20) to state the algorithms for solving the three composite types is, for me, a strong argument for advancing another hypothesis. Indeed, I think that we are in a case comparable to the twelfth-century mathematical treatise Liber Mahameleth [Book on transactions] of which Jacques Sesiano writes:

The Liber mahameleth thus clearly relies on Arabic sources, and this is hardly surprising for the time and place of its writing. But we may even affirm that the Liber mahameleth was (initially) written in an Arabic environment. Indeed, its reader was supposed to have, or to be able to have, access to some Arabic treatises and read them in the original. (Sesiano 2014, xviii).

That is to say that the Liber restauracionis would not be a book that was originally in Arabic. It would instead be an original Latin text written in the context of the Arabic algebra with the specific contributions of the Islamic West. Indeed, besides the complete absence of Arabisms in the Liber restauracionis, I think that it is philologically risky to claim that the versified stanzas, at least, could have been translated into Latin from an Arabic composition. Hence, the author would be a mathematician from al-Andalus or from the Iberian Peninsula more broadly. Whether or not he is an Arabist, he would write his text directly in Latin by knowing or directly consulting Arabic texts produced in the Islamic World and probably also the Arab-Latin translations available (at least that of Gerard of Cremona). This author is mathematically educated and particularly well informed about literature then extant, especially with regard to Euclid's Elements (Books II, VII and X in particular).

Finally, considering the didactic characteristics of Liber restauracionis that we have highlighted (particularly versification, brevity and a systematically analytical presentation), I also hypothesize that it would have been written to spread Algebra in the environements of Latin students and scholars from Mediaeval Europe, discovering the sciences of the Islamic countries. If so, and if I still had to designate a possible role for Guglielmo of Lunis, then I could only consider him as the translator of a Latin-Italian version of which (U) would be a witness.

Critical Edition

Critical Apparatus

In addition to the references to the three manuscripts -(F), (L) and (V) -I give the pagination of the Kaunzner's edition (K) as, for example: /p. 50/. For convenience, I followed [START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF]) by considering (L) as the main manuscript. I nevertheless considered, if necessary, the best readings between (F), (L) and (V) and I corrected some Kaunzner's transcription errors. All alterations and variants between the three copies, even minor, are taken into account in my critical apparatus. As I wanted it as simple as possible, I use only few necessary abbreviations.

om. omitted (and not added otherwise by the scribe) ad. added in a marginal or interlinear gloss cor. corrected by the original scribe For example, vel] om. F; means: the word vel is omitted in (F). quod] ad. LV; means: the word quod is added in both (L) and (V) by the scribe.

dragmas] dragmis cor. dragmas F; means: the word dragmis is corrected by dragmas in (F). Sometimes, the correction is highlighted by a sign as here. I preserved as much as possible the signs of correction of the scribes as pointed letters, crossed out words and other underlined words (in particular in dotted line). Most of the time, these signs refer to corrections made by the scribe himself. Bold letters are used for section titles (only present in the margins of (V)) to make the text easier to read. I added also punctuation in order to enhance comprehensibility.

In my edition, all geometric letters are written in italic capital letters. In the manuscripts, they are in lowercase and framed by dots. In (L), they are also easily recognized by the lines drawn over them. For example, the square ABGD is noted "tetragonus .abgd." in (F) and (V), "tetragonus .abgd." in (L).

The writing of numbers is different between the three manuscripts and even within the same text. I specify in footnotes the different representations chosen by scribes: literal, Hindu-Arabic or Roman numerals (in two cases).

For the geometrical diagrams, I chose to reproduce all of them for each copy. They are often different because they were not necessarily completed in all the copies. In addition, I indicate the main differences between the three copies in footnotes, including the absence of diagrams in one or the other of the copies. I also number the figures to make reference easier.

Finally, I respected the orthography of the three scribes, including -unlike [START_REF] Kaunzner | Die lateinishe Algebra in MS Lyell 52 der Bodleian Library, Oxford, früher MS Admont 612[END_REF]) -the classical mediaeval diphthongs (que for quae among others). Orthographic distinctions between scribes (like doubling of letters) are included in footnotes. For the abbreviations of mediaeval Latin, I consulted to the classical Dizionario di abbreviature latine ed Italiane [START_REF] Cappelli | Dizionario di abbreviature latine ed italiane[END_REF]. In some cases, I keep the text of the manuscript even if the written word has no meaning for me; in that case, I inscribe (?) after this word. 52 qui ex ducta radice in se producitur] qui ex dicta radice in se ducta producitur F; qui ex dicta radice in se ducta procreatur V;

53 collatione] collactione V;

54 collata] collocata multiplicabo in 10 169 dragmas, que sunt latus BE, unde proveniat superficies AE; ex quo igitur 10 radices censui, una cum dragmis 21, coequantur. Census autem est tetragonus AG, relinquitur superficies DE /p. 61/ quantum dragme 21. Cum itaque dividitur per medium linea BE ad punctum Z, erit 170 hoc 171 inter puncta G E aut inter puncta B G.

73r.

169 10] decem V;

170 erit] erit ad. cadet V;

171 hoc] ergo V;

Fit 172 hoc prius inter puncta B G, 173 ergo per 5 tam secundi elementorum 174 tetragonus 175 BZ equatur 176 producto ex 177 BG in GE, una cum adiuncto sibi tetragono ZG. Duco igitur dimidias radices, que sunt BZ, in se, et fit tetragonus 178 25, cui subtraho productum ex BG in GE, quod est 21, et remanentis tetragoni radicem sumo, 179 scilicet 4, velud GZ 180 linea, qua 181 adiuncta dimidiis radicibus, id est BZ, impleta est [F, fol.61va] BG, radix quesiti census. /p. 62/ Si autem punctum Z inter 182 puncta G E locabitur, 183 necesse est, ut 184 linea, que modo dimidiis radicibus coacervata radicem census implevit, ex eisdem sublata, radicem census in reliquo patefaciat, et hoc fuit probandum.

[V, fol.73v] Hic probatur qualiter radices et numerus equantur censui. 185 Tercii vero modi, qui est: 3 186 radices et 4 187 dragme censum equant. Ratio haec est: ponam tetragonum 188 ABGD censum, 189 cuius radicem AB in 3 190 dragmas implico, 191 que sint latus BE, unde proveniat super [L, fol.44v]ficies AE tribus radicibus equa; 192 quod vero reliquum est tetragoni, scilicet super-

172 Fit] Sit V; 173 Fit hoc prius inter puncta B G] om. F; 174 5 tam secundi elementorum] 5 secundi elementorum L; "per quintam secundi euclidis" in marginalia, different hand (V);

175 tetragonus] tetragus ad. onus V;

176 equatur] equantur V; 177 equatur producto ex] equatur ex producto ex F;

178 tetragonus] tetragus V; 179 remanentis tetragoni radicem sumo] remanent tetragoni radicum summo F; remanentis tetragoni radicem id est sumo L;

180 GZ] GE L;

181 qua] que V;

182 inter] ut FL; intus V; 2x 2 -3x; 2x 2 -4; 5x -2x 2 ; 5x - Deinceps vero 341 in accipiendo cuiuslibet radicis quocumque 342 multiplice, censum eius in 343 tetragonum numeri, a quo multiplex denominatur, multiplica, et provenientis summe radix propositam radicem [F, fol.62rb] se-cundum illum numerum multiplicat. Verbi gratia. Si radicis 344 9 duplicem sumere volueris, 9 345 in tetragonum binarii, 346 a quo duplex denominatur, multiplica, et efficies 36, cuius radix, scilicet 6, proposite radicis duplex est.

183 puncta G E locabitur] puncta B G F; puncta G E
Si autem radicis alicuius quamcumque 347 submultiplicem 348 quesieris, censum eius per tetragonum numeri, a quo submultiplex denominatur, divide, et exeuntis summe radix proposite radici 349 secundum illum numerum submultiplex erit. Verbi gratia. [L, fol.46v Secunda questio, imposita in secundo minorum, ubi census equatur numero. 418 [#II] Dividimus 419 itaque 420 10 in tales partes, quarum unius tetragonus 421 in duas dragmas et 7 nonas ductus tetragono tocius equivaleat. Quarum inven [F, fol.62vb]tionis regula haec est. 422 Sit AB, ut consuevimus, 10, AC vero 402 consequens] continens V; 403 The two following geometric representations are only in (V). In (U) -the Latin-Italian translation -, and only in (U), there is a third diagram.

404 sit] fit V;

405 res] om. Questio imposita in primo capitulo maiorum. 445 sus et 100 minus 20 rebus efficiunt, [F, fol.63va]

Figure 1 :

 1 Figure 1: Viable stemma of the manuscripts of the Liber restauracionis

F

  , fol.61r L, fol.43v (Read E instead of D, at the bottom right corner) and L. fol.44r V, fol.73r

Figure 2 :

 2 Figure 2: Geometrical diagrams for the procedure associated with x 2 + 10x = 39.

F

Figure 4 :

 4 Figure 4: Geometrical diagrams for the procedure associated with 3x + 4 = x 2 .

Figure 5 :

 5 Figure 5: From left to right: F, fol.61v (reference sign at the top left); L, fol.44v; V, fol.73v

Figure 6 :

 6 Figure 6: Symbolic format for +2x 2 ; +3x; +4

Figure 7 : 4 5

 74 Figure 7: From left to right: F, fol.61v (reference sign at the top left); L, fol.44v; V, fol.73v

Figure 10 :

 10 Figure 10: From left to right: F. fol.61v (reference sign at the top left); L. fol.44v; V. fol.73v

Figure 11 :

 11 Figure 11: Symbolic format for2x 2 -3x; 2x 2 -4; 5x -2x 2 ; 5x -4

Figure 12 :

 12 Figure 12: From left to right: F. fol.62r; L. fol.46r; V. fol.74r; modern transcription

Figure 13 :

 13 Figure 13: From left to right: F. fol.62r; L. fol.46r; V. fol.74v; modern transcription

Figure 14 :

 14 Figure 14: V. fol.75r; modern transcription

Figure 15 :

 15 Figure 15: V. fol.75r; modern transcription

Table 1 :

 1 Table of common readings: quantitative approach

Table 2 :

 2 Comparison of geometric diagrams and tables of numbers in manuscripts (the number mentioned is that of my edition)

  4.2. Latin Text/p. 50/[V, fol.72r] Incipit liber, qui secundum arabes vocatur algebra et almucabala, et apud nos liber restauracionis nominatur, et fuit translatus a magistro Giurardo cremonense in toleto de arabico in latinum. 23[L, fol.42r] [F, fol.60va] Unitas est principium numeri et non est numerus. 24 Numerus enim 25 est collectio unitatum; sicut ergo omnis numerus multiplex unitatis est, 26 sic eadem multiplex est ab eo denominate sue partis. Verbi gratia: ut ternarius triplex est unitatis, sic unitas tripla est 27 sue tercie partis, a ternario denominate; unde necesse est, ut quemadmodum infinitas multiplicationes suscipit unitas, infinitas quoque partes eadem intellectu et ratione suscipiat. Omnis autem numerus non incongrue infra 9 28 terminos artatur, 29 ut ex decuplatorum eorumdem iteratione innumerabilis 30 multitudo profluat, 31 scilicet ut habita naturali numerorum dispositione, a prima unitate usque ad 9 32 terminos fiat progressus. Et hii quidem primo 33 limiti attribuantur. Consequenter quolibet terminorum decuplato 34 alii 9 35 procreantur, qui secundum limitem custodiant, scilicet 10, 20, 30 et caetera usque in 36 90. Hii quoque ad eundem modum decuplati 37 in tercio limite collocentur, ut 100, 200, 300 et sic usque ad 38 900. Ad quorum similitudinem infinitos limites 39 novenis terminis dispositos generandi regula teneatur. Ad omnem compotum perutilis nichil est enim 40 computationi subserviens praeter numerum. 41 Incipiuntur capitula algebre et primo de tribus minoribus. 42 Capitulum primum minorum. 43 Qualiter census equatur radicibus. 44 Numerus autem, secundum quod 45 nostre computationi necessarius est, in 3 a dividitur. 46 In 47 radicem numeri, et quadratum sive censum radicis, et numerum simplicem, nec censui nec radici comparatum. Radix autem est numerus 48 , quo 49 in se ducto alter procreatur. 50 Census autem vel 51 quadratus Incipiuntur capitula algebre et primo de tribus minoribus] ad. V; om. FL; 43 Capitulum primum minorum] Capitulum primum minorem ad. V; om. FL; 44 Qualiter census equatur radicibus] ad. V; om. FL; 45 quod] ad. V; 46 in 3 a dividitur] in trina dividitur F; et in trina dividitur V; radicis est numerus, qui ex ducta radice in se producitur. 52 Numerus simplex est is, qui per se producitur, nulla censum /p. 52/ vel radicem collatione 53 respiciens. Haec tria singula singulis collata, 54 3 55 modos constituunt, scilicet ut census equus fit aliquot 56 radicibus, vel census numero, vel 57 radices numero. Verbi gratia. Census radicibus, quotiens 58 dicitur: census equatur 5 radicibus, unde constat quod [L, fol.42v] radix illius est 5 et census est 59 25, is enim suis 5 radicibus adequatur. Vel cum dicitur: tercia [F, fol.60vb] census adequatur 4 60 radicibus, ergo 61 census implet 12 radices, et hic est 144. Vel sic 5 62 census equantur 63 10 radicibus, unus igitur census duabus radicibus equipollet. 64 Palam ergo quoniam radix est 65 2, 66 census eius 4. In talibus quoque id observari 67 convenit, ut, quotiens 68 plus minusve censu alteruter reliquorum 69 comparatur, ad integritatem unius census convertatur, converso pariter reliquo sub simili cremento vel diminutione. Capitulum secundum minorum.

	29 artatur] arteratur V;
	30 innumerabilis] internumerabilis L;
	31 profluat] perfluat V;
	32 9] novem FV;
	33 primo] primi FV;
	34 terminorum decuplato] decuplatorum terminorum L;
	35 9] novem FV;
	36 et caetera usque in] usque FL;
	37 modum decuplati] decuplati modum V;
	38 et sic usque ad] et caetera usque in V;
	39 infinitos limites] infinitos unitas limites L;
	40 nichil est enim] nichil est F; nichil enim est V;
	41 praeter numerum] nisi numerus V;
	42 47 In] Scilicet in F;
	48 est numerus] numerus est F;
	49 quo] qui V;
	50 procreatur] producitur FV;
	51 vel] om. F;

70

Qualiter census equatur numero. 71 Sed census 72 numero confertur hoc modo: 73 census equus est 9, et est 9.

  Vel 5 census valent 80, ergo unus census valet quintam partem de 80, 74 et hoc est 16. Vel medietas census valet 18, ergo ipse est 36.Capitulum tercium minorum.75 Qualiter radices equantur numero.76 Radices vero numero equivalent 77 hoc modo: radix valet 3, ergo est 3. Vel 4 78 radices equipollent 20, una /p. 53/ igitur est 5. Vel si medietas ra[V, fol.72v]dicis valet 10, radix est 20.Hic incipitur de tribus capitulis maioribus.79 Ex his 80 tribus simplicibus 81 tres permixti generantur, si quelibet bina tercio conferantur: 82 census et radices numero, census et numerus radicibus, radices et numerus 83 censui.Qualiter census et radices equantur numero.84 Census et radices numero, quotiens dicitur: 85 Census et 10 radices equivalent 39 unitatibus, per quod intelligendum 86 est quidam census, cui si copulentur 10 87 radices sui, erit totum hoc 39 dragme. 88 Ad cuius scientiam haec regula 89 perducit. 74 Vel 5 census valent 80, ergo unus census valet quintam partem de 80, et hoc est 16] Vel 5 census valent 80, ergo unus census valet quintam partem 80, hoc est 16 F; Vel 5 census valent 80, ergo unus census valet quintam partem 80, et hoc est 16 L; Vel 5 census valent 40, ergo unus census valet quintam partem de 40, hoc 8 V; Qualiter radices equantur numero. Radices vero numero equivalent hoc modo: radix valet 3, ergo est 3. Vel 4 radices equipollent 20, una igitur est 5. Vel si medietas radicis valet 10, radix est 20. Hic incipitur de tribus capitulis maioribus.] om. F; ad. Super radices que numeros quoequantur et ostendit aliud allgebra (?) ad hoc signum (and followed by a circle)F; Capitulum primum. 90 Cum rebus censum siquis dragmis 91 dabit 92 equum, Res quadra medias, quadratis 93 adice dragmas, Radici quorum medias res excipe demum, Et residuum quaesiti census 94 radicem ostendet. Verbi gratia. Radices in proposita quaestione sunt 10, 95 harum as/p. 54/ sumptam medietatem, id est 96 5, quadra, hoc est in se multiplica footnote hoc est in se multiplica] om. V; et erunt 25. Adice his 97 dragmas, 98 quae sunt 39, et ef[L, fol.43r]ficies 64, ex quorum accepta 99 radice, scilicet 8, dimidias radices diminue, 100 unde relinquntur 101 3, quod est radix, census 102 vero est 103 9. Quotiens 104 autem plus minusve censu 105 in quaestione proponitur, ad unum censum converte; converso pariter reliquorum utroque secundum census crementum vel diminucionem. Verbi gratia. Si proponantur 2 census et 10 radices 106 equivalentes dragmis 48, 107 per conversionem unus census et 5 108 radices equivalebunt dragmis 24. 109 Vel si medietas census et 5 110 radices equantur 111 12, per conversionem [F, fol.61ra] integer census et 10 radices equabuntur 112 24. Qualiter census et numerus equantur radicibus. 113 Census autem et numerus equipollent radicibus, quotiens 114 dicitur: census et 21 dragme 115 equipollent 10 116 radicibus, per quod intelligendus 117 est quidam census, cui si copulentur 118 21, erit totum hoc equale 10 119 radicibus eiusdem census. Ad cuius noticiam hac regula perduceris. Capitulum secundum. 120 Cum censu dragmas siquis rebus dabit equas, Res quadra medias, quadratis abice 121 dragmas, /p. 56/ Dimidiis rebus reliqui latus adde vel aufer, Et exiens quesiti census radicem ostendet. Verbi gratia. Radices in proposita questione sunt 10, harum medietatem quadra, hoc est 5, et efficies 25. Ex his 122 diminue dragmas, que sunt 21, et relinquntur 123 4, cuius acceptam radicem, scilicet 2, dimidiis radicibus subtrahe vel adde, subtrahe restabunt 3, 124 quod est radix, census eius 9; vel adde et habebis 7, quod est radix, census eius 49. Experire igitur hanc questionem tum addendo tum subtrahendo, quoniam per alteram certificaberis. Hic etiam nota: si quadrate dimidie radices propositis dragmis pauci[L, fol.43v]ores 125 fuerint, questionem esse impossibilem; si vero equales, erit radix census equa 126 dimidiis radicibus. In huius modi quoque 127 quociens 128 plus minusve censu proponuntur, singula convertendo reduc in 129 unum censum. Qualiter radices et numerus equantur censui. 130 Radices vero et numerus equantur censui, quotiens 131 dicitur: 3 radices et 4 dragme equivalent censui, 132 quod 133 hac regula comprehendes. 134 [V, fol.73r] Capitulum tercium. 135 Si census rebus dragmisque requiritur equus, Res quadra medias, quadratis adice 136 dragmas, Quorum radicem mediis radicibus adde, Et collectum quesiti census radicem indicabit. 137 Verbi gratia. Radices in proposita questione sunt 3, 138 harum assumptam medietatem quadra, et habebis 2 139 et quartam. His adice 140 dragmas, scilicet 4, et efficies 6 141 et quartam, horum radicem, /p. 57/ scilicet duo et semis 142 , dimidiis radicibus adde, unde colligentur 4, quod est radix, census autem est 143 16. Predictorum 6 144 modorum 3 145 priores adeo evidentes sunt, quod probatione non egent. 146 Posteriores [F, fol.61rb] vero ex secundo elementorum sue necessitatis firmitatem trahunt, quorum primi -Hic probatur qualiter census et radices equantur numero -147 quidem, quod est: 148 census et 10 radices equivalent dragmis 39. Ratio hec est: ponam tetrago-num ABGD censum; 149 eius 150 radicem DG multiplicabo in 10 dragmas, que sint linea GE 151 directe continuata linee BG, unde proveniat superficies DE, ex quo igitur census cum 10 radicibus equatur dragmis 39. Necesse est 152 superficiem quoque 153 AE 39 dragmas 154 implere. Divisa itaque per medium linea 155 GE ad punctum Z, cui directe coniuncta 156 est linea BG, per 6 tam /p. 58/ secundi elementorum 157 fit productum ex EB in GB, qui est superficies AE dragmarum 158 39, una cum 159 adiuncto sibi tetragono GZ, equa 160 te[L, fol.44r] 161 tragono BZ. Duco igitur dimidias radices, que sunt 5, in se, unde fit tetragonus GZ, 162 cui addo multiplicationem EB in GB, que est 39 dragme, et colliguntur 64, ex quorum sumpta radice, que est BZ, diminuo dimidias radices, 163 id est GZ, et quod relinquitur est GB, 164 radix scilicet census propositi. /p. 60/ Probacio qualiter census et numerus equantur radicibus. 165 Ratio vero secundi, quod est: 166 census et dragme 21 167 equantur 10 radicibus. Talis est: ponam censum tetragonum ABGD, 168 cuius radicem AB

	90 Capitulum primum.] om. FL;
	91 siquis dragmis] dragmis siquis F;
	F; 56 fit aliquot] sit aliquibus V; 55 3] tres FV; 57 vel] a vel V; 58 quotiens] quociens F; 59 est] om. FL; 60 4] quatuor V; 61 ergo] igitur V; 62 5] quinque V; 63 equantur] adequantur V; 64 equipollet] equippollet L; 65 Palam ergo quoniam radix est] Palam ergo est quoniam radix est F; 66 2] duo V; 67 observari] obscurari V; 68 quotiens] quociens F; 69 alteruter reliquorum] alteruter reliquorum scilicet radix et numerus F; 70 Capitulum secundum minorum] Capitulum secundum minorem ad. V; om. FL; 71 Qualiter census equatur numero] ad. V; om. FL; 72 Sed census] Census vero V; 73 hoc modo] hoc ut F; 80 his] hiis FL; 81 simplicibus] principibus V; 82 conferantur] referantur FV; 83 numerus] numeri V; 84 Qualiter census et radices equantur numero.] om. FL; 85 Census et radices numero, quotiens dicitur] census et radices numero equantur quo-ciens dicitur F; Census et radices numero, est hac si dicat V; 86 intelligendum] intelligendus FL; 87 10] decem V; 88 dragme] dragma L; 89 haec regula] om. L; 92 dabit] dabis V; 131 quotiens] quociens F; 93 quadratis] quadratum V; 132 3 radices et 4 dragme equivalent censui] Tres radices et quatuor dragme equantur 94 census] ad. F; 95 10] decem V; 96 id est] scilicet V; 97 his] hiis FL; 98 dragmas] dragmis cor. dragmas F; 99 accepta] assumpta F; 100 dimidias radices diminue] dimidias radicas radices diminue F; dimidias res minue V; 101 relinquntur] reliquetur V; 102 census] et census V; 103 est] om. FL; 104 Quotiens] Quociens F; 105 censu] om. F; 106 2 census et 10 radicem] duo census et 10 radicem F; duo census et decem radicem V; 107 dragmis 48] 48 dragmis F; 108 5] quinque F; 109 equivalebunt dragmis 24] equabuntur 24 dragmis V; 110 5] 5 cor. 2 1 2 F (in marginalia, other hand?); 112 equabuntur] equivalebunt V; 129 in] ad V; 111 equantur] equentur V; censui V; 113 Qualiter census et numerus equantur radicibus.] om. FL; 133 quod] quem V; 114 quotiens] quociens F; 134 comprehendes] deprehenderis V; 115 dragme] dragma FL; 135 Capitulum tercium.] om. FL; 116 10] decem V; 136 adice] addice L; 117 intelligendus] intelligendum V; 137 indicabit] ostendet V; 118 copulentur] copuletur FV; 138 3] tres V; 119 10] decem V; 139 2] duo V; 120 Capitulum secundum.] om. FL; 140 His adice] Hiis adice F; Hiis addice L; 121 abice] addice L; 141 6] sex V; 122 his] hiis FL; 142 duo et semis] 2 et semis F; 123 relinquntur] relinquetur V; 143 est] om. FL; 124 subtrahe vel adde subtrahe, restabunt 3] subtrahe vel adde, restabunt 3 L; subtrahe 144 6] sex V; vel adde, restabunt 3 V; 145 3] tres V; 125 pauciores] pociores F; 146 egent] indigent V; 126 equa] eque V; 147 Hic probatur qualiter census et radices equantur numero] om. FL; 127 In huius modi quoque] In huius quoque F; Hi quoque huius L; 128 quociens] quotiens V; 148 quod est] quod qui est F; quod cum V;

75

Capitulum tercium minorum] Capitulum tercium minorem ad. V; om. FL; 76 Qualiter radices equantur numero] ad. V; om. FL; 77 equivalent] equipollent V; 78 4] quatuor V; 79 Vel medietas census valet 18, ergo ipse est 36. Capitulum tercium minorum.

130

Qualiter radices et numerus equantur censui.] om. FL;

  loquabitur cor. ca V;184 ut] quod F; 185 Hic probatur qualiter radices et numerus equantur censui.] om. FL; Geometrical diagrams for the procedure associated with x 2 + 21 = 10x.ficies ED, dragmis 4 equipollet. 193 Deinde divisa 194 linea BE per medium 195 ad punctum Z, cui directe continuata est EG, fit productum 196 ex BG in GE, una cum tetragono ZE, equum tetragono ZG per sextam secundi euclidis. 197 Duco igitur ZE, id est dimidium radicum, in se, et fit 2 et quarta, 198 cui aggrega 199 productum ex BG in GE, id est superficiem ED, que est 4, et colligo 6 200 et quartam, quorum sumo radicem, scilicet ZG, que est 2 201 et semis, qua 202 dimidio radicum, hoc est BZ, coacervata, fit /p. 63/ propositi census radicem BG perficio, cuius figura est hec. 203 Qualiter figurentur census, radices et dragme. 204 Porro omnis compotus, qui in restauratione diminuti vel projectione superhabundantis 205 exercetur, ad aliquod horum 6 206 capitulorum convertibilis est, quod, ut levius fiat discenti, quedam scribendi et multiplicandi praecepta damus, quibus integre 207 res ad invicem, 208 necnon res, quibus diminuitur vel superhabundat numerus, aut quae diminuuntur 209 vel superhabundant numero, multiplicentur, hoc praesumpto 210 , quod ex ductu rei in rem provenit tantum census, et ex ductu rei in numerum non nisi rerum multitudo. In scribendo quidem 211 haec regula teneatur: numero censuum littera c, numero radicum littera r, deorsum virgulas habentes, subterius apponantur, dragme vero sine litteris virgulas habeant, quotiens 212 haec sine diminutione praeponuntur. 213 Verbi gratia: 2 214 census, /p. 64/ 3 215 radices et 216 4 dragme sic figurentur (Fig. 5 & 6).

	F, fol.61r
	L, fol.44r
	V, fol.73r
	Figure 3: 193 equipollet] equippollet L; 186 3] tres FV; 194 Deinde divisa] Divisa deinde FV; 187 4] quatuor V; 195 per medium] om. F; 188 tetragonum] tetragum cor. onum V; 196 fit productum] fit per 6 secundi elementorum productum F; 189 This diagram is introduced in the text. The diagrams in (F) and (L) are obviously 197 per sextam secundi euclidis] om. FL; "Ista probantur per sextam secundi euclidis", wrong, ABGD is clearly not a square. "per sextam secundi euclidis" in marginalia, different hand (V); 190 3] tres FV; 191 implico] multiplico FV; 198 2 et quarta] duo et quarta V;
	192 equa] equat F;

  4 ipsum, quod in multiplicatione 232 /p. 66/ digitorum et articulorum observamus, ut res sit tamquam articulus 233 superhabundans vero vel diminutum tamquam digitus. Si enim proponantur 234 digitus et articulus in digitum 235 et articulum multiplicandi, quadripartimur totam multiplicationem per 4 secundi elementorum, 236 scilicet ut digitus 237 in digitum, 238 articulus in articulum subalterne, digitus in articulum, articulus in digitum, contraditorie multiplicetur, 239 ut diminutus, si diminutum [V, fol.74r] multiplicet, 240 vel superhabundans superhabundans; 241 inde productum multiplicationi 242 reliquorum coacervate 243 totius proposite multiplicationis summam ostendit. Verbi gratia. Proponantur 10 et 2 244 multiplicanda in 10 et 3. Ducimus 245 2 246 in 3 et erunt 247 6, item 10 in 10 et erunt 100, item 248 2 in 10 et erunt 20, item 249 10 in 3 et erunt 30, quorum summa 156. 250 Proponantur quoque 10 minus duobus 251 in 10 minus tribus 252 ducenda. Multiplicamus 2 diminuta in 3 diminuta et erunt 6 addenda, item 253 10 in 10 et erunt 100, item 254 duo 255 diminuta in 10 et erunt 20 minuenda, item 256 10 in 3 257 diminuta et erunt 30 minuenda, 258 ergo reliquorum summa 56 ex multiplicatione provenit. Qualiter census, radices et dragme inter se multiplicentur. 259 Ad instar horum multiplicamus res, quibus diminuitur vel superhabundat 260 numerus vel a 261 ipse numero. Verbi gratia. Proponantur 262 10 dragme minus duabus 263 rebus multi[L, fol.45v]plicande 264 in 10 265 dragmas minus tribus rebus. Multiplicamus duas 266 res diminutas in 3 267 res diminutas 268 et erunt 6 census addendi, item 10 dragmas in 10 fiunt 100, 269 item 270 2 271 res diminutas in 10 dragmas fiunt 20 res minuende, item 272 10 in 3 res diminutas 273 fiunt 30 274 res minuende. Summa igitur huius multiplicationis est 6 census et 100 dragme minus 50 rebus. Item multiplicentur 10 dragme plus duabus rebus in 10 dragmas minus tribus 275 rebus. 2 res addite 276 in 3 res diminutas fiunt /p. 67/ 6 census 277 minuendi, 10 in 10 fiunt 100 dragme, 2 res addite in 10 fiunt 20 res addite, 278 10 in 3 res diminutas fiunt 30 279 res minuende. 280 Restauramus 281 ergo per 20 priores res res 30 282 diminutas, et sic erunt tantum 10 283 res diminute. Ex hac ergo multiplicatione surgunt 284 100 dragme minus 6 censibus et 10 rebus. Similiter agendum est 285 in multiplicatione numerorum cum fractionibus propositorum. [F, fol.62ra] Proponantur enim due 286 dragme et tercia, multiplicande in 2 287 dragmas et unam quartam. Tercia in quartam fit 12 a , 288 item 289 2 in 2 290 fiunt 4, item 291 tercia in 2 fiunt 2 e tercie, 292 item 293 2 in quartam 294 fit semis. Totum ergo fit 5 dragme et 3 duodecime. 295 BD, per que restauretur radix 200 rum , 319 ergo AD est ple[V, fol.74v]na radix 200 rum , 320 DC vero reliquum est 10 minus radice 200 rum321 . Ergo tota AC est radix 200 rum322 et insuper 10 minus eadem radice, quam diminutam, 323 loco eiusdem superhabundantis 324 abicimus, et relinquitur AC 10.

  rum , 328 quod reliquum est equivalet 30 minus duabus radicibus 200 rum . 329 Verbi gratia. Sit AB 20 minus radice 200 rum , 330 cui subtrahatur AC, que sit radix 200 rum minus 10. 331 Dico ergo 332 quod reliquum CB est 30 minus duabus 333 radicibus 200 rum . 334 Applicabimus igitur 335 ad AB 10, que sint DA. Palam est ergo quod tota DB 336 est 30 minus radice 200 rum , 337 ex qua sublata DC, hoc est plena radice 200 rum , 338 relinquitur CB 30 minus duabus radicibus 200 rum . 339

	319 200 rum ] 200 V;
	320 200 rum ] 200 V;
	321 10 minus radice 200rum] minus 10 radice 200rum F; 10 minus radice de 200 V;
	322 radix 200 rum ] radix 200rum L; radix de 200 V;
	323 diminutam] diminuta V;
	324 superhabundantis] superhabundans F;
	325 Hic eciam tractatur de radicibus.] om. FL;
	326 200 rum ] 200 V; 200rum L;
	327 20 minus] 20 minus minus F;
	328 200 rum ] 200 V; 200rum L;
	329 radicibus 200 rum ] radicibus 200 V; radicibus 200rum L;
	330 Sit AB 20 minus radice 200 rum ] Sit AB 20 minus radice 200 V; Sit AB minus radice
	20 200rum L;

  ] Si radicis 36 subduplum inquiris, 36 per tetragonum binarii, a quo denominatur subduplus, 350 divide, et exibunt 9, quorum radix proposite radici subdupla 351 est. Hiis memoriter observatis, quaslibet numerorum radices, notas seu surdas, 352 duplicare, triplicare, dimidiare, vel terciare, facillimum est. Quotiens 353 autem radi/p. 71/cem census unius in radicem alterius ducere placuerit, censum per censum multiplica, et provenientis summe radix est quantum provenit ex multiplicatione propositarum radicum. Cum vero quemlibet duplicem 354 radicis unius in quemlibet multiplicem radicis 355 alterius ducere volueris, produc censum utriusque multiplicationis, 356 unum per alterum multiplicans, unde provenientis summe radix dabit summam, que ex ductu radicis multiplicis unius in alterius multiplicatione 357 alterius 358 colligetur. Verbi gratia. Si duplicem radicis 9 in triplicem radicis 359 4 ducere placuerit, produc censum duplicis radicis 360 9, hoc est 36, et censum tripli-consequens 402 est ergo, ut reliqua CB sit 10 minus re. 403 Porro ducatur res in se et sit 404 census, eadem quoque res 405 ducatur in 10 minus re et sunt 406 10 res minus censu, quarum quadruplum esse censum, quem 407 res ex se protulit, necesse est. Mox igitur, ut in quatuor 408 multiplicate fuerint, 10 res minus censu 409 producantur 410 40 res minus 4 censibus, uni censui equales. Restaurabimus ergo diminuta, idque aliter fieri nequit, 411 [L, fol.47v] nisi 4 census diminuti 412 tam radicibus quam censui accedant. Quo facto, integre 40 res 5 censibus equivalent, quibus, ad simplicem censum reductis, erit unus census 8 413 rebus equivalens. 414 Palam ergo, quod res una est 8 415 , scilicet pars AC, relinquitur itaque CB 2, et sic in primum 6 minorum, 416 qui est: census equivalet 417 radicibus, quaestio nostra soluta est.

  V; 406 sunt] fiunt V; 407 quem] quam V; 408 quatuor] 4 FV; 409 10 res minus censu] om. FL; 410 producantur] producentur FV; 411 nequit] nequed V; 412 diminuti] diminute F; diminue . te L; 413 8] scilicet F; 414 8 rebus equivalens] 8 rebus equivales L; equivalens 8 rebus V; 415 8] om. F; 416 minorum] 4 or F; 4 L; 417 equivalet] equivalens V; 418 Secunda questio, imposita in secundo minorum, ubi census equatur numero.] om.

	FL;
	419 Dividimus] Dividamus V;
	420 itaque] om. FV;
	421 tetragonus] tetragus LV;
	422 haec est] est haec F;

  quibus 4, 529 ut supradictum est, equivalet necessario unius per alteram multiplicatio, ducta 530 in ea, que ex 531 utriusque per reliquam divisionem proveniunt. Hoc vero datum est 2 a et sexta. 532 Duc ergo rem in 10 minus re et habebis 10 res minus censu, que in 2 et sextam multiplica, unde collecta 21 res 533 et 2 tercias rei minus duobus censibus et sexta census confer 534 tetragonorum coniunctioni, hoc est 2 bus535 censibus et 100 minus 20 rebus. Si ergo 536 utraque creverint equaliter, compensatis scilicet diminutis censibus, fiunt 21 res et 2 537 tercie rei equales 100 et 4 censibus et 6 te538 census minus 20 rebus. Et post restaurationem rerum 41 res et 2 539 tercie rei equivalebunt 100 [L, fol.49v] et 4 censibus et 540 6 te quoque census, 541 que necesse est, ad unum censum converti. Considerandus est 542 ergo, quota pars vel partes sit unus 543 census 4 censuum et 6 te census; 544 est autem 5 ta et 5 te partis 5 ta , 545 et tantumdem 546 dragmarum et rerum ad invicem 547 conferatur. Ergo unus census cum 24 equivalet 10 rebus; resolve per 5 tum548 modum, 549 et res erit 4.

Incipit liber, qui secundum arabes vocatur algebra et almucabala, et apud nos liber restauracionis nominatur, et fuit translatus a magistro Giurardo cremonense in toleto de arabico in latinum.(ms. Vat. Lat. fol.72r)

Una traduzione dall' arabo in latino, fatta in Toledo da Gherardo cremonese d'un importante trattato d'algebra, trovasi monoscritta nel codice Vaticano n • 4606 dal recto della carta 72 al rovesco della carta 76. Questa versione essendo inedita, parmi utile di riportarla qui appresso colla traduzione in linguaggio algebrico d' alcune regole, ed operazioni che in essa sono indicate.(Boncompagni 1851, 27-

28)3 Charles Burnett does not mention it in(Burnett

2001).4 "La maggior parte del contenuto di questo codice, fatta eccezione per il contenuto delle carte dalla 81 alla 102, è una traduzione in volgare di parti del manoscritto latino Vat. lat. 4606 (sec. XIV) della Biblioteca Apostolica Vaticana."(Franci

2003, 31) 5 Description of the manuscript in(Van Egmond 1980, 215-217). Warren Van Egmond specifies in particular that "this ms. is largely an Italian trans. of the Latin text in Vat. Lat. 4606, which in turn is heavily based on Leonardo Pisano's Liber abbaci ".

Note that the scribe used the Roman numeral twice: x for 10, iiii for 4.

An exception in fol.60v with a 2 1 2 in the margin. I conjecture that it is from another hand.

One single initial is not inserted (fol.62v); we can conclude this from the reserved space provided for this purpose and the tiny letter in marginalia.

I sincerely thank the historians of sciences David Juste and Henry Zepeda and the palaeographer Marc Smith for their help in understanding these glosses.

For this reason, I will give it in its entirety in the edition.

As David Juste confirmed this understanding, I will not propose here an edition of this note.

David Juste confirms this hypothesis in a personal communication.

We thus have the differences of one manuscript with the other two.

"There are three Latin translations: by Gerard of Cremona, by Robert of Chester and by William of Luna. On the translation by Gerard of Cremona, which is by far the best and the most literal, see (...) On the translation by Robert de Chester, see (...)"[START_REF] Rashed | Al-Khwārizmī: The beginnings of Algebra[END_REF], 86, English translation from Rashed 2007, 86) Nothing is said about the aforesaid third translation.

A segnare il passaggio di questo metodo alla cultura europea, stanno le tre traduzioni latine realizzate fra il XII e il XIII secolo in Spagna, l da Roberto di Chester (1145), Gerardo da Cremona (1170) e da Guglielmo de Lunis (1250) (...)(Ambrosetti 2011, 138)

-10) + (20 -√ 200) = 10 and (20 -√ 200) -( √ 200 -10) = 30 -2 √ 200 (p.32-34);

Incipit liber, qui secundum arabes vocatur algebra et almucabala, et apud nos liber restauracionis nominatur, et fuit translatus a magistro Giurardo cremonense in toleto de arabico in latinum] om. FL;

et non est numerus] om. L;

enim] om. F;

unitatis est] est unitatis F;

tripla est] est tripla V;

9] novem V;

aggrega] aggregabo F; agrega V;

6] sex V;

2] duo V;

qua] quam V;

perficio cuius figura est hec] proficio figura hec est. I ad tale figuram (I is a sign referring to the diagram in marginalia) F; proficio cuius figura hec est V;

radix 200 rum minus 10] radix 200 rum minus x F; radix minus 10 V; radix 200rum minus 10 L;

Appendix A. The problems in the Liber restauracionis: Modern mathematical transcription

We have two series of problems in the Liber restauracionis. For easy reference, I have numbered the problems with #j (1 ≤ j ≤ 6) in Roman numerals for the first one, and #i (1 ≤ i ≤ 10) in Arabic numerals for the second one. In the first series, each type of linear and quadratic equations is represented in the same order that their algorithms of resolution were enunciated in the previous part. In the second series, the organization of the problems do not seem to meet any particular requirement.

First series:

#I Divide 10 into two parts x and y in such a way that x + y = 10 and x 2 = 4x × y. #II Divide 10 into two parts x and y in such a way that x + y = 10

(2 + 7 9 )x 2 = 10 2 . #III Divide 10 into two parts x and y in such a way that x + y = 10 and All the problems of the first series are identical in the Mukhtas . ar of al-Khwārizmī (Rashed 2007, 144-156;Rashed 2009, 144-156), in the two wellknown Arabic-Latin translations ( (Hughes 1986, 217) for Gerard and (Hughes 1989, 53-56) for Robert) and in the Liber restauracionis. Nevertheless, we have to mention the difference for #I and #II with geometrical representations in the Liber restauracionis, absent from the other texts (for more details (Moyon 2017, 247-249)).

For the second series, I reproduce here the table of comparison established in (Moyon 2017, 268) with a correction for #10.

584 I give here the correct statement of the ninth problem in relation to (Moyon 2017, 267). Taking into account the use of the term census (māl in al-Khwārizmī's text) as the square of the root, one could have written the problem as follows: "Determine a square number y in such a way that (y -(


Mayuri Shukla

Subhamita Maitra

Jean-François Hernandez

Piyarat Govitrapong

Bruno Vincent
email: bruno.vin@mahidol.ac.th

Methamphetamine regulates βAPP processing in human neuroblastoma cells

Keywords: Alzheimer's disease, methamphetamine, drug abuse, βAPP, secretase, metabolism

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

It has long been accepted that long term methamphetamine (Meth) abuse (high repetitive doses) produces neurotoxicity and is associated with cognitive impairments [START_REF] Ares-Santos | Neurobiology of methamphetamine[END_REF], the neuropsychological deficits being associated with the neurodegenerative effects of this drug as observed in experimental models [START_REF] Cadet | Speed kills: cellular and molecular bases of methamphetamine-induced nerve terminal degeneration and neuronal apoptosis[END_REF][START_REF] Krasnova | Methamphetamine toxicity and messengers of death[END_REF]. Several mechanisms of Meth-induced neurotoxicity have been proposed, including oxidative stress, excitotoxicity, mitochondrial dysfunction and neuroinflammation marked by microgliosis, astrogliosis and cytokine induction [START_REF] Moszczynska | Molecular, behavioral, and psychological consequences of methamphetamine neurotoxicity: Implications for treatment[END_REF], concurring to mediate apoptosis and neurotoxicity in the CNS [START_REF] Sharikova | Methamphetamine induces apoptosis of microglia via the intrinsic mitochondrial-dependent pathway[END_REF]. Beside perturbations in calcium and lipid homeostasis, post Meth exposure is characterized by degeneration of neurons, similar to what is observed in Alzheimer's disease (AD) pathology.

Indeed, a neuroproteomic study carried out in rats has shown that the expression of 18 proteins (11 in the hippocampus and 7 in the olfactory bulb) underwent a significant alteration as a result of Meth exposure with the altered proteins being involved in cell death, inflammation, oxidation, and apoptosis [START_REF] Zhu | The effect of chronic methamphetamine exposure on the hippocampal and olfactory bulb neuroproteomes of rats[END_REF]. Interestingly, all these pathogenic mechanisms are part of the neurodegenerative cascade of Alzheimer's disease (AD) [START_REF] Morales | Neuroinflammation in the pathogenesis of Alzheimer's disease. A rational framework for the search of novel therapeutic approaches[END_REF][START_REF] Kocahan | Mechanisms of Alzheimer's disease pathogenesis and prevention: The brain, neural pathology, N-methyl-D-aspartate receptors, Tau protein and other risk factors[END_REF] and involvement of the limbic system and hippocampus in particular exemplifies the connection between Meth abuse and AD pathology.

It has also been demonstrated that the GSK3β kinase as well as the pleiotropic transcription factor NFκB, which respectively play key roles in Tau phosphorylation and in inflammatory responses, both participate to the control of Aβ42-induced inhibition of ADAM10 expression and augmentation of BACE1 and presenilin 1 transcription [START_REF] Buggia-Prevot | NFkappaB-dependent control of BACE1 promoter transactivation by Abeta42[END_REF][START_REF] Chami | Nuclear factor-κB regulates βAPP and βand γ-secretases differently at physiological and supraphysiological Aβ concentrations[END_REF][START_REF] Chinchalongporn | Melatonin ameliorates Aβ42induced alteration of βAPP-processing secretases via the melatonin receptor through the Pin1/GSK3β/NF-κB pathway in SH-SY5Y cells[END_REF]. Interestingly, Meth is capable to induce pro-inflammatory cytokines and their mediators through the NFκB pathway [START_REF] Permpoonputtana | The anti-inflammatory effect of melatonin on methamphetamine-induced proinflammatory mediators in human neuroblastoma dopamine SH-SY5Y cell lines[END_REF] and to promote TNFα expression as well as NFκB nuclear translocation [START_REF] Wongprayoon | Melatonin attenuates methamphetamine-induced neuroinflammation through the melatonin receptor in the SH-SY5Y cell line[END_REF]. Given the additional fact that Tau phosphorylation is induced by 3,4-Methylenedioxymethamphetamine ("Ecstasy") in vivo in the mouse hippocampus [START_REF] Busceti | Enhanced Tau phosphorylation in the hippocampus of mice treated with 3,4-Methylenedioxymethamphetamine ("Ecstasy")[END_REF], these mechanisms altogether portray that Meth-induced alterations correlate with AD-like pathology.

Nevertheless, although substantial number of evidences validates the pernicious effects of Meth on the CNS, other studies have contrastingly ascertained some neuroprotective effects of this psychostimulant [START_REF] Rau | Administration of low dose methamphetamine 12 h after a 16 severe traumatic brain injury prevents neurological dysfunction and cognitive impairment in rats[END_REF][START_REF] Rau | The neuroprotective potential of low-dose methamphetamine in preclinical models of stroke and traumatic brain injury[END_REF] and the genuine effect of Meth abuse on cognition is still matter to debate [START_REF] Hart | Is cognitive functioning impaired in methamphetamine users? A critical review[END_REF]. Therefore, Meth presents a challenging ambiguity of neurotoxicity and neuroprotection where the potential underlying mechanisms are precisely regulated by the controlled exposure of Meth dosages. Given the neurodegenerative effects and its sodality with the cognitive functioning, which majorly depends on dosage variation and time of exposure, we reasoned that Meth administration might differentially affect the proteolytic processing of βAPP, a field of investigation that has been relatively unexplored so far.

Herein, we investigated the effect of various doses of methamphetamine on βAPP metabolism and βAPP-cleaving secretases in vitro in cultured human SH-SY5Y neuroblastoma cells. With emphasis on the benefits and hazards of Meth on cognitive, functional and behavioral manifestations, the idea is to develop a better understanding of the drug action and disease process and the possibilities of modification of the AD pathogenesis and treatment.

Materials and methods

Materials

DMEM complete medium, Opti-MEM, trypsin and fetal bovine serum (FBS) were from Invitrogen (Carlsbad, CA, USA). Penicillin-streptomycin mix was from PAA. Poly-D-lysine was from Sigma (St. Louis, MO, USA). Tris buffer and glycine were from Vivantis. Skim milk powder was from criterion. ECL and ammonium persulphate were from GE Health care (Piscataway, NJ, USA). SDS was from Amresco. O-Phenanthroline was from Calbiochem (San Diego, CA, USA). (D)-METH hydrochloride was purchased from Alltech-Applied Science (State College, PA, USA).

Cell culture and treatments

Human neuroblastoma SH-SY5Y cells were cultured at 37°C, 5% CO2 in DMEM supplemented with 10% inactivated FBS containing penicillin (100 U/ml) and streptomycin (50 mg/ml). For experiments, the cells were seeded in six-35mm well plates and were grown until reaching 80% confluence. Cells were then treated without (control) or with various concentrations of Meth (1, 10, 100 and 1000 μM) for 16 h.

Western blot analysis in cell lysate

Cells were collected with phosphate-buffered saline (PBS)-EDTA and resuspended in 80-150 μl of lysis buffer (10 mM Tris/HCl, pH 7.5, 150 mM NaCl, 0.5% triton X-100, 0.5% deoxycholate, 5 mM EDTA). Protein concentrations were determined by the Bradford method [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF] and 5-50 µg of proteins were separated by SDS-polyacrylamide gel electrophoresis on 8% (βAPP), 10% (ADAM10 and BACE1) or 12% (β-actin) Tris/glycine gels. Proteins were then transferred onto nitrocellulose membranes (45-120 min according to protein size, 100 V), blocked for 1 h in 5% nonfat milk and incubated overnight at 4°C with primary antibodies (in 5% nonfat milk) directed against βAPP (dilution 1/4000, polyclonal A8717, Sigma), ADAM10 (dilution 1/500, polyclonal AB19026, Millipore), BACE1 (diution 1/1000, monoclonal Ab108394, Abcam) or β-actin (dilution 1/5000, monoclonal 13E5; Cell Signaling) antibodies. Bound antibodies were detected using goat anti-mouse (dilution 1/3000, polyclonal 7076, Cell Signaling) or goat anti-rabbit peroxidase-conjugated (dilution 1/3000, polyclonal 7074, Cell Signaling) antibodies. Immunological complexes were revealed using Immobilon Western Chemiluminescence HRP substrate (Millipore) and detected using an automatic medical X-ray processor (Kodak, Rochester, NY, USA). Films were scanned, band densities were measured with the Image J software (http://imagej.nih.gov/ij/), and data were normalized using β-actin as an internal standard.

Measurement of sAPPα secretion

SHSY5Y cells were cultured in 35-mm dishes until they reached 80% confluence. Then, the cells were treated without (control) or with various concentration of Meth in DMEM containing 1% FBS. Media were then removed and replaced with 1 ml of serum-free DMEM and cells were allowed to secrete for 3 h. Then, TCA precipitation of the 1 ml serum-free secretion media was performed and samples were subjected to Western blot analysis onto 8% SDS-PAGE, transferred onto nitrocellulose membrane (120 min, 100 volts), incubated in 5% non-fat milk blocking solution for 30 min and incubated overnight at 4°C with the humanspecific anti-sAPPα antibody DE2B4 (dilution 1/500, monoclonal DE2B4, Santa Cruz Biotechnology Inc., Santa Cruz, CA, USA). Membranes were then incubated with HRPconjugated anti mouse antibody and processed for quantification as described above.

α-secretase fluorimetric assay on intact cells

SH-SY5Y cells cultured in 35-mm dishes coated with polylysine (10 μg/ml) were treated without (control) or with Meth for 16 hours and the α-secretase catalytic activity was measured using the JMV2770 substrate and the ADAM10-specific GI254023X inhibitor as previously described [START_REF] Alfa Cisse | Design and characterization of a novel cellular prion-derived quenched fluorimetric substrate of α-secretase[END_REF].

β-secretase fluorimetric assay of cell homogenates

SH-SY5Y cell lysate (30 µg) were treated without (control) or with Meth for 16 hours and assayed for their β-secretase activity using the JMV2236 substrate and the JMV1197 BACE1-specific inhibitor as previously described [START_REF] Andrau | BACE1-and BACE2-expressing human cells: characterization of beta amyloid precursor proteinderived catabolites, design of a novel fluorimetric assay, and identification of new in vitro inhibitors[END_REF].

Real-time quantitative polymerase chain reaction (q-PCR)

SH-SY5Y cells were treated in the absence (control) or in the presence of various Meth concentrations as described above. Post treatment, total RNA was extracted and purified with the PureLink RNA mini kit (Ambion, Life Technologies, Austin, TX, USA). Real-time PCR was performed with 100 ng of total RNA using the QuantiFast SYBR Green RT-PCR kit (Qiagen, Singapore) detector system (Eppendorf Mastercycler ep RealPlex) and the SYBR Green detection protocol. The 2x QuantiFast SYBR Green RT-PCR master mix, QuantiFast RT mix, QuantiTectPrimer Assay and template RNA were mixed and the reaction volume was adjusted to 25 μl using RNase-free water. The specific primers were designed and purchased from Qiagen. Each primer is a 10x QuantiTect Primer Assay containing a mix of forward and reverse primers for specific targets: Hs_ADAM10_1_SG (QT00032641) (human ADAM10), Hs_BACE1_1_SG (QT00084777) (human BACE1) and Hs_GAPDH_1_SG (QT00079247, human GAPDH, housekeeping gene for normalization).

Statistical analysis

Statistical analyses were performed with the GraphPad Prism software (San Diego, CA, USA) using the unpaired t-test for pair wise comparisons.

Results

Effect of methamphetamine on the α-secretase processing of βAPP

We first investigated the effect of Meth on the non amyloidogenic α-secretase processing of endogenously expressed βAPP in cultured human SH-SY5Y neuroblastoma cells by measuring the production of the neurotrophic, neuroprotective, memory-enhancing and neurogenesis-stimulating βAPP-derived sAPPα metabolite. Our results interestingly showed that Meth dose-dependently increases sAPPα secretion when applied at 1 and 10 µM, while sAPPα production was not changed when further increasing Meth concentrations up to 100 µM and 1 mM (Fig. 1). Moreover, none of the Meth doses applied altered βAPP immunoreactivity (Fig. 1), thereby indicating that Meth is genuinely controlling the αsecretase processing of βAPP rather than altering its expression or maturation.

Effect of methamphetamine on a-secretase catalytic activity

We then studied the effect of different concentrations of Meth on the catalytic activity of the principal α-secretase ADAM10 in wild-type SH-SY5Y by means of an α-secretasespecific fluorimetric assay. The results show that, as was observed with sAPPα production, 1 and 10 μM of Meth significantly and dose-dependently increase the GI254023X-sensitive hydrolysis of the JMV2770 substrate when compared to untreated cells (Fig. 2A) whereas higher doses of the psychoactive drug (up to 1 mM) failed to modify ADAM10 activity (Fig. 2A).

Effect of methamphetamine on ADAM10 transcription and protein expression

To determine whether the observed up regulation of ADAM10 catalytic activity is a consequence of a Meth-dependent transcriptional activation of the protease, we measured the effect of Meth on ADAM10 mRNA by quantitative real-time PCR experiments. The results depict a similar pattern of action of Meth on ADAM10 mRNA expression when compared to the one previously observed for sAPPα production and ADAM10 catalytic activity, with a dose-dependent stimulation at 1 and 10 µM and no effect at 100 µM and 1 mM (Fig. 2B) and a similar pattern was observed when assessing the impact of the drug on ADAM10 immunoreactivity/protein levels by western blot (Fig. 2C).

Effect of methamphetamine on BACE1 catalytic activity and expression

We have then investigated the effects of Meth on the β-secretase BACE1, which is the rate-limiting initiator of Aβ production. Indeed, following incubation of SH-SY5Y cells with various concentrations of Meth for 16 h, we have evidenced a dose-dependent increase of BACE1 catalytic activity with a significant difference observed only at 1 mM when compared to untreated control cells (Fig. 3A). In a second set of experiments, we have established by quantitative real-time PCR that the drug indeed increases BACE1 mRNA levels when applied at the high 100 µM and 1 mM concentrations but not at the low 1 and 10 µM doses, thereby indicating that high Meth can control BACE1 at a transcriptional level (Fig. 3B). Finally, western blot analysis performed under the same conditions have established that Meth treatments positively control BACE1 protein levels (Fig. 3C). The a priori paradoxical observation that Meth at 100 µM did not show any effect on β-secretase catalytic activity while significantly increasing BACE1 mRNA levels could be due to the presence of Meth-insensitive non-BACE1 β-secretase activities able to cleave the fluorimetric substrate, whereas the quantitative PCR is fully specific for BACE1 and certainly displays a higher sensitivity when compared to the fluorimetric assay.

Discussion

The present demonstration that Meth could convey beneficial effect regarding Alzheimer's disease is in line with previous reports having established that psychostimulants can be used medically to fight brain disorders [START_REF] Iversen | Speed, Ecstasy, Ritalin: The Science of Amphetamines[END_REF]. Indeed, amphetamine and Meth are the first line treatment for attention deficit hyperactivity disorder (ADHD) [START_REF] Surman | Do stimulants improve functioning in adults with ADHD? A review of the literature[END_REF] as well as for the management of narcolepsy [START_REF] De La Herran-Arita | Current and emerging options for the drug treatment of narcolepsy[END_REF]. Furthermore, clinical trials have provided evidence that Meth may be effective in the treatment of otherwise treatment-resistant anxiety disorders [START_REF] Mithoefer | The safety and efficacy of +/-3,4-methylenedioxymethamphetamine-assisted psychotherapy in subjects with chronic, treatment-resistant posttraumatic stress disorder: the first randomized controlled pilot study[END_REF] and that its administration is involved in supportive and integrative psychotherapy protocol [START_REF] Danforth | MDMA-assisted therapy: A new treatment model for social anxiety in autistic adults[END_REF].

More specifically considering brain functions, a significant amount of data has evidenced beneficial effects of this psychoactive drug under correct dosage and right timings/duration of exposure. Regarding physiological status, juvenile rats administered with Meth showed improved performance in a spatial navigation task when tested at adulthood [START_REF] Mcfadden | The effects of methamphetamine exposure during preadolescence on male and female rats in the water maze[END_REF]. In addition, Meth controls the release of the catecholamine dopamine that are capable of activating multiple neuroprotective pathways in the brain [START_REF] Rau | Low dose methamphetamine mediates neuroprotection through a PI3K-AKT pathway[END_REF]. Now considering pathological conditions, it has been demonstrated that low to moderate doses of Meth induces short-term improvements in learning and memory in preclinical models of stroke and traumatic brain injury [START_REF] Rau | The neuroprotective potential of low-dose methamphetamine in preclinical models of stroke and traumatic brain injury[END_REF].

In this context, the present study strongly suggests that the above mentioned beneficial effects of low Meth doses can, at least partly, be due to the enhancement of α-secretase expression and activity leading to an increased production of sAPPα, which is a wellestablished neuroprotective and neurotrophic factor, but is also a key contributor to synaptic plasticity and spatial memory and has been more recently shown to enhance neurogenesis [START_REF] Mockett | Therapeutic potential of secreted amyloid precursor protein APPsα[END_REF].

When briefly considering the possible mechanisms through which Meth could promote sAPPα production, it is important to firstly underline here that Meth, after its binding to neuronal dopamine receptors, can activate the ERK pathway [START_REF] Mizoguchi | Regulations of methamphetamine reward by extracellular signal-regulated kinase 1/2/ets-like gene-1 signaling pathway via the activation of dopamine receptors[END_REF] and the phosphorylation of the CREB transcription factor [START_REF] Liu | Effect of rhynchophylline on the expression of p-CREB and sc-Fos in striatum and hippocampal CA1 area of methamphetamine-induced conditioned place preference rats[END_REF]. Because CREB, a downstream target of ERK, is involved in ADAM10 promoter transactivation [START_REF] Shukla | Melatonin stimulates the nonamyloidogenic processing of βAPP through the positive transcriptional regulation of ADAM10 and ADAM17[END_REF][START_REF] Meng | Tat-haFGF14-154 upregulates ADAM10 to attenuate the Alzheimer phenotype of APP/PS1 mice through the PI3K-CREB-IRE1α/XBP1 pathway[END_REF], low Meth could positively control ADAM10 expression via the CREB pathway, the incapacity of higher concentrations of the drug to operate in the same manner being due to an desensitization of the receptors. Secondly, because Meth increases BDNF and TrkB in multiple brain regions and since BDNF promotes the non-amyloidogenic processing of βAPP and reduces the production of Aβ peptides in a transgenic mouse model of AD [START_REF] Nigam | Exercise and BDNF reduce Aβ production by enhancing α-secretase processing of APP[END_REF], BDNF might play an intermediate role in the Methdependent sAPPα augmentation. Thirdly, because Meth induces glutamate release causing over activation of NMDAR and AMPAR receptors in the brain [START_REF] Simões | Methamphetamine changes NMDA and AMPA glutamate receptor subunit levels in the rat striatum and frontal cortex[END_REF] and since synaptic NMDAR receptor activation stimulates α-secretase processing of βAPP and inhibits Aβ production [START_REF] Hoey | Synaptic NMDA receptor activation stimulates alpha-secretase amyloid precursor protein processing and inhibits amyloidbeta production[END_REF], it could be speculated that adjusting the dosage paradigm of Meth would be another channel to regulate and modulate NMDAR receptor activation. Finally, the key regulatory role of calcium in cognition, together with the Ca 2+ -dependent nature of αsecretase [START_REF] Chen | The maze of APP processing in Alzheimer's disease: Where did we go wrong in reasoning? Front[END_REF], might provide an additional route through which Meth could possibly influence βAPP processing by α-secretase.

Although disruption of BACE1 has been recently shown to affect amphetamine-dependent dopaminergic signalling in the midbrain, an area implicated in schizophreniform behaviours [START_REF] Paredes | Physiological and behavioral effects of amphetamine in BACE1 -/mice[END_REF], the present study is the first one to report an effect of Meth on BACE1 and the here described Meth-induced up-regulation of BACE1 expression could possibly be under the control of several molecular mechanisms. Firstly, it has been shown that chronic Meth administration increases α-synuclein protein levels in the hippocampus [START_REF] Butler | Chronic methamphetamine increases alpha-synuclein protein levels in the striatum and hippocampus but not in the cortex of juvenile mice[END_REF]. Since αsynuclein expression induces BACE1 protein levels [START_REF] Roberts | α-Synuclein increases β-amyloid secretion by promoting β-/γ-secretase processing of APP[END_REF], it could therefore be inferred that the high Meth-induced rise in BACE1 levels seen in our present study might use α-synuclein as an intermediate factor. Secondly, Meth administration can cause neurotoxicity and neurodegeneration via the up regulation and nuclear translocation of NFκB [START_REF] Wongprayoon | Melatonin attenuates methamphetamine-induced neuroinflammation through the melatonin receptor in the SH-SY5Y cell line[END_REF] and by increasing GSK3β and tau phosphorylation in a dose-and time-dependent manner [START_REF] Li | Role of GSK3β/α-synuclein axis in methamphetamine-induced neurotoxicity in PC12 cells[END_REF], thereby triggering apoptosis in a GSK3β-dependent manner [START_REF] He | Amphetamine neurotoxicity in PC12 cells through the PP2A/AKT/GSK3β pathway[END_REF]. Because NFκB and GSK3β have an impact on βAPP metabolism via the modulation of BACE1 expression [START_REF] Chinchalongporn | Melatonin ameliorates Aβ42induced alteration of βAPP-processing secretases via the melatonin receptor through the Pin1/GSK3β/NF-κB pathway in SH-SY5Y cells[END_REF], it can therefore be speculated that our observed effect of high Meth treatment on BACE1 might involve both NFκB and GSK3β. Thirdly, initially reported as the most instantaneous effect of moderate to high dose of Meth is the production of ROS and oxidative stress [START_REF] Yang | The main molecular mechanisms underlying methamphetamineinduced neurotoxicity and implications for pharmacological treatment[END_REF]. Considering this parameter, Meth-induced oxidative stress has been shown to lead to a dysfunctional processing of βAPP in rat and human platelets in AD patients [START_REF] Ehrlich | Effects of oxidative stress on amyloid precursor protein processing in rat and human platelets[END_REF] with enhanced secretion of sAPPβ [START_REF] Muche | Oxidative stress affects processing of amyloid precursor protein in vascular endothelial cells[END_REF] that further increases BACE1 processing of βAPP via BACE1 gene activation [START_REF] Tan | Effects of mild and severe oxidative stress on BACE1 expression and APP amyloidogenic processing[END_REF]. Finally, it is well established that mitochondrial dysfunction affects βAPP expression and processing as well as Aβ accumulation [START_REF] Swerdlow | The Alzheimer's disease mitochondrial cascade hypothesis: Progress and perspectives[END_REF]. Moreover, some metabolic disturbances seen in AD likely arise from increased ER-mitochondrial communication that is driven by an increase in the levels of C99, the C-terminal processing product of βAPP derived from its cleavage by βsecretase that is present in mitochondria-associated endoplasmic reticulum membranes [START_REF] Area-Gomez | A key role for MAM in mediating mitochondrial dysfunction in Alzheimer disease[END_REF]. In this context, the fact that Meth mediates ER stress leading to apoptosis [START_REF] Shah | Methamphetamine-mediated endoplasmic reticulum (ER) stress induces type-1 programmed cell death in astrocytes via ATF6, IRE1α and PERK pathways[END_REF] could provide another mechanism through which Meth could control the BACE1 cleavage of βAPP.

Altogether, our data shed light on possible mechanisms through which Meth could convey both beneficial and deleterious effects regarding AD through differential effects on βAPP processing, depending on the concentrations applied. Thus, one can first delineate a so-called "therapeutic" window (1-10 µM) in which sAPPα production is at the pic of the bell-shape curve and BACE1 is not yet activated (Fig. 4, green area), thereby directing the metabolism of βAPP towards its amyloidogenic pathway. Secondly, one enter an AD-promoting window in which higher Meth concentrations (100 µM-1 mM) do not have an effect on sAPPα secretion anymore but dose-dependently increase BACE1 expression and catalytic activity, thereby shifting βAPP processing toward the amyloidogenic pathway (Fig. 4, red area). It will now be of particular interest to determine whether Meth could also have an impact on γsecretase.

Considering the possible therapeutic use of Meth in AD, it is worth noting that this compound has a relatively long half-life and crosses the blood brain barrier rapidly and that low doses of Meth produce very few side effects and the fact that Meth is metabolized to amphetamine, further prolongs its activity in the brain [START_REF] Cruickshank | A review of the clinical pharmacology of methamphetamine[END_REF][START_REF] Panenka | Methamphetamine use: A comprehensive review of molecular, preclinical and clinical findings[END_REF]. For these reasons, Meth has significant potential as a neuroprotective agent and could therefore be possibly used as a substitute to manipulate the cognitive outcomes in AD patients under strict supervision. In order to further characterize the pharmacology of Meth as a potential anti-AD agent and define possible parameters of clinical application, the precise therapeutic window in which it would be possible to intervene should now be thoroughly examined in vivo in animal models of the disease. and memory-enhancing and neurogenesis-stimulating sAPPα fragment (green). On the other hand is the AD-promoting window (100 µM-1 mM), which might correspond to drug abusing condition and that no longer augments sAPPα production but instead increases BACE1 expression and activity (red).

Figure 1 :

 1 Figure 1: Effect of methamphetamine on the α-secretase processing of βAPP in cultured

Figure 2 :

 2 Figure 2: Effect of methamphetamine on α-secretase catalytic activity and ADAM10

Figure 3 :

 3 Figure 3: Effect of methamphetamine on β-secretase catalytic activity and BACE1

Figure 4 :

 4 Figure 4: Schematic representation of the concentration-dependent beneficial and deleterious effects of methamphetamine on βAPP processing. This diagram illustrates the two antagonistic windows of action of methamphetamine regarding βAPP processing by the nonamyloidogenic α-secretase and the amyloidogenic β-secretase activities. On one hand is

Acknowledgements

This work was supported by Mahidol University, NSTDA (National Science and Technology Development Agency) Discovery based Development grant (P-11-00783) and The Thailand Research Fund (IRG 5780009, RSA5880042 and BRG6180002). SM is supported by a Mahidol University postdoctoral fellowship.