Dominique Raynaud

Pietro Cavallini

Giovanni Di Muro Della Marca

G Vasari

Geometrical and Arithmetical Methods in Early Medieval Perspective

Keywords: Middle Ages, perspective, painting, arithmetic, geometry. of Isaac, sometimes identified with Arnolfo di Cambio 49, 92 mm = 1 o., 59, 90 mm = 2

This paper examines the hypothesis that early perspective paintings were drawn arithmetically, without vanishing points. The best argument for this hypothesis is that the division of two parallel lines by straight lines intersecting each other at the vanishing point (geometrical method) is equivalent to the division of those parallel lines in proportional parts (arithmetical method). If arithmetical method had been used, then the vanishing points exhibited ex post should be purely fortuitous. But the lack of multiples and submultiples of measurement units, the absence of proportionality ratios, the length of the operating series, and the correspondence of vanishing points to visible loci of the picture offer sound objections for this hypothesis. The use of optics and geometrical method is more probative-though it does not mean that painters were using concepts of linear perspective, which would be an anachronism.

INTRODUCTION

Perspective is a legitimate object for the history of science when painting layouts obey a mathematical scheme. In spite of the alleged contributions by Brunelleschi and Alberti, the history of perspective as a science begins only in Piero della Francesca's De prospectiva pingendi (1470) 1 . The present paper will try to apply methods in the history of science to the duecento and trecento, as well as to more distant periods. This is a tentative essay calling for further research, and we deliberately focus on a situation that is poorly documented: the frescoes painted on the lower register of the Upper Church in Assisi. Here are the reasons justifying this choice: 1° Contemporaneous documents concerning these mural paintings are practically nonexistent. The names of the painters that were working on the site are only assumed. As for many ancient large-scale projects, numerous hands have been suggested: Jacopo Torriti, Cimabue, the Master of the Capture, the Master of Isaac, Giotto, etc. Take the particular case of The Legend of St Francis scenes. The spectrum of attributions ranges from Giotto di Bondone 2 and the Master

• PLC, Université Pierre-Mendès-France (Grenoble), GEMAS (Paris). 1 J.V. FIELD, Alberti, the abacus and Piero della Francesca's proof of perspective, «Renaissance Studies», 11, 1997, pp. 61-88. 2 L. BELLOSI, G. RAGIONIERI, Giotto e le storie di San Francesco nella basilica superiore du Assisi, in Assisi anno 1300, a cura di S. Brufani e E. Menestò, Assisi, Edizioni Porziuncola, 2002, pp. 455-473. drawn with a compass; curves being directly drawn on the coating; and faces being transferred by means of drawings used as a kind of template (patroni). In general, preliminary drawings were not kept. 10 The frescoes of the Upper Church at Assisi are known only scarcely: the crumbling coating enables us to discover the reddish brown pigment underdrawing (sinopia) on the left side of St Francis Preaching before Pope Honorius III; 11 the examination in low-angled light makes it possible to see the sign of the line in the fresh coating, the perforations of the nails maintaining the line and, at times, the painter's fingerprints, as in the Extasis. 12 The double transfer of the disegno to the sinopia, and then onto the intonaco, poses a difficult problem to the history of perspective. The layout, required from the earliest stage, was partly disappearing in the double transfer. Seen in low-angled light, the vanishing lines in perspective are generally limited to the edges of the buildings. 13 4° Documentary gaps preclude a direct apprehension of the knowledge that was underlying the perspective layouts. That is why these mural paintings have given rise to multifarious interpretations, encompassing the whole spectrum from intellectualism to empiricism. There is no best way. Intellectualism runs the risk of anachronism by accrediting concepts of horizon, vanishing point, etc., that were undefined at that time, whereas empiricism encounters an underevaluation of the overt or tacit knowledge that is necessary to draw in perspective. And yet the frescoes of the Upper Church at Assisi are unquestionably pioneering trials in the rationalization of the visual space. These frescoes thus await an interpretation able to reconcile empirical evidence with conceptual minimalism. 14

THE ARITHMETICAL CONSTRUCTION HYPOTHESIS

In order to explain the duecento and trecento improvement in the expression of depth, historians first supposed that artists already had at their disposal a basic knowledge of convergence, 9 Pittura a fresco. Tecniche esecutive, cause di degrado, restauro, a cura di G. Basile, Firenze, Le Monnier, 1989; B. ZANARDI, C. FRUGONI, F. ZERI, Il Cantiere di Giotto. Le storie di San Francesco ad Assisi, Milano, Skira, 1996. 10 B. DEGENHART, A. SCHMITT, Einleitung a Corpus der Italienischen Zeichnungen 1300-1450, I-1. Sud-und Mittelitalien, Berlin, Gebr. Mann Verlag, 1968, p. XIX. 11 "[…] Subito sopra l'aureola du Francesco è visibile una piccolissima traccia, del tutto indecifrabile, d'un disegno in rosso sinopia. Ma, al di là di questa minima prova è l'enorme complessità iconografica del ciclo a rendere impossibile l'ipotesi di una esecuzione a fresco delle scene in assenza d'un detagliatissimo progetto su carta (o pergamena) da riportare al vere sull'arriccio in forma di sinopia […] Questi dati materiali impongono di nuovo di dar per certo un assai dettagliato lavoro di progettazione del ciclo francescano, prima con disegni su carta (o pergamena) e poi sull'arriccio", B. ZANARDI, Il cantiere di Giotto, op. cit., pp. 24, 32. 12 Ibid., pp. 24-32. 13 infinity, etc. This interpretation 15 has given rise to well-known objections: (1) Mathematical knowledge that was at the disposal of craftsmen is poorly known, and mostly hypothetical; (2)

The convergence on a defined vanishing point is only occasional: we observe more often a convergence on a "vanishing axis" or on a less accurately defined "vanishing region"; (3) Trials in the rationalization of pictorial space have led to many competing systems, hence reducing the importance of linear perspective as such. Such difficulties invalidate much of the credibility granted to Panofsky's opinion that painters such as Lorenzetti had a mathematical consciousness of the vanishing point, thought of as a "symbol of infinity." 16 Empiricism is the opposite of intellectualism. Among the empiricist stances, that of Andrés de Mesa Gisbert demandes attention because it combines rigor with minimalism. The author proposed a hypothesis to explain the convergence of vanishing lines in fourteenth-century painting.17 This is in fact a method for building a perspective arithmetically. "Si disponemos dos rectas paralelas con cualquier distancia entre sí, y luego de dividir una de ellas en un número cualquiera de partes lo hacemos en forma similar sobre la segunda paralela, guardando exactamente las mismas proporciones con las que se lo ha hecho inicialmente, al unir los puntos correspondientes con líneas rectas, en su prolongación obtendremos la convergencia de todas ellas sobre un solo y único punto sin necesidad de haber operado con él" (Ibid., p. 33, italics mine). The author's basic insight is that convergence of segments AD, BE, CF… may be obtained either by drawing the lines OA, OB, OC… whose segments AD, BE, CF… are the visible parts (geometrical method), or by proportioning segments DE, EF… to segments AB, BC…, the proportionality ratio between AB and DE, BC and EF being sufficient to ensure the existence of the virtual vanishing point O (arithmetical method). Among the relations on similar triangles, craftsmen would have extracted relation (1), leaving aside relations (2), (3), etc., which make the concurrent point O intervene (Ibid., pp. 33-34, Figure 1). The arithmetical hypothesis explains quite simply why ex post layouts show a point at infinity: it would be a by-product of the analysis that presupposes the use of geometrical devices.

But, in fact, painters could have used a proportionality rule to divide the frontal-horizontal lines and distribute them in depth.

In addition, de Mesa's hypothesis solves three problems affecting pre-renaissance paintings: (1) it accounts for the arbitrary behavior of certain lines, invoking that proportional ratios were not defined on all lines (observe parallel edges of the abaci of the capitals in St Francis

Preaching before Pope Honorius III);

(2) it explains by way of the same reason the additional presence of a vanishing axis as in The Pentecost; and (3) the convergence on a vanishing region is seen as a secondary effect of the errors made during the transfer of proportional segments (any inaccurate positioning of points A, B, C will induce a deviation of vanishing lines AD, BE, CF); this arithmetical scheme is often cited to emphasize anachronism in the attempts to recognize the beginnings of linear perspective in medieval times. 18 Despite its obvious ingenuity and usefulness, the arithmetical method gives rise to difficulties that were never systematically explored. This is understandable: when a conclusion seems correct, rarely do we thoroughly question its premises. The arithmetical hypothesis is highly liable: nothing proves that medieval craftsmen made use of vanishing points. Until now the arithmetical method has been hypothetical, not factual. There is a need for further scrutiny.

LACK OF MULTIPLES OR SUBMULTIPLES OF MEASUREMENT UNITS

In a perspective painting composed according to the arithmetical method, frontal lines ought to be divided into multiples or submultiples of common measurement units. The choice of a set of measures is never arbitrary, whether having a symbolic or practical value. The history of ar-chitecture provides many illustrations of this. Take for example the alleged geometric layout of the cappella Pazzi by Brunelleschi. 19 Konrad Hecht 20 inaugurated the critical approach by paying attention to the discrepancy of seventeen regulator layouts published between 1867 and 1957. In the same vein, Jean Guillaume 21 has shown that the regulator layouts imagined to explain the architectural composition do not match well with the Cappella dei Pazzi measurements, to which Brunelleschi always gave integer or simple values-e.g. pilasters are 1

€ 1 2 braccia in width.
Transposed to the case of mural painting, the discovery of units multiples would be a good clue that the arithmetical method had been used: (1) it is easier to calculate proportional ratios on simple measurements; and (2) the means is favorable with regard to memory and communication. Do early perspective paintings exhibit multiples or submultiples of measurement units?

In Umbria, the two competing systems used were the braccio and the piede. As in many other regions, there were two values for the braccio: the braccio corto (0,599 m) and the braccio lungo (0,668 m) documented in Perugia, Foligno, Orvieto, Spoleto, etc. Choose the second one, inasmuch as the braccio corto was called "da legname" and "da muratori", whereas the braccio lungo was specified as a unit "da lana", "da panno" or "da seta". This choice established the system of measures: braccio (599 mm), oncia (49,92 mm), soldo (29,95 mm). The Umbrian value for the piede "da legname e da fabbrica" was 0,363 m, from which we can deduce a second system of measures: piede (363 mm), palmo (90,75 mm), pollice (30,25 mm), and dito 14) presumes to be an exemplary application of arithmetical method. Draw the layout of this fresco, 23 and report the main measurements on it. Mark on the layout: above the line on the left side, the coffer radius; below the line, the coffer diameter; above the line on the right side (underlined), the total length of the vaulted ceiling (Figure 2).

(
---------------------------------------------

Piede-palmi system

Horizontal dim. (mm) The dimensions of the coffers can hardly be expressed in one or another system of units.

Vertical dim. (mm) ---------------------------------------------
The measurements corresponding to integer values ± 3 mm are 2 out of 11 in the braccio-soldi system, 3 out of 11 in the piede-palmi system. This fact raises a doubt on the existence of an underlying system of measurements.

Apply now the same analytic method to the fresco of St Francis Preaching before Pope Honorius III, which represents a groined-vault roofed space.24 The main measurements of this space can hardly be expressed in one or another systems of units. The dimensions close to an integer value ± 3 mm are 3 out of 19 in the braccio-soldi system, 7 out of 19 in the piede-palmi system.

---------------------------------------------

Piede-palmi system

Horizontal dim. (mm) Vertical dim. (mm) ---------------------------------------------
Consider now The Recovery of the Wounded Man of Lerida, whose flat-coffered ceiling is a checkered pattern in perspective. 25 On the perspective scheme (Figure 4): the first number on the left side represents the width of the narrowest coffer of the row; the second one, the coffer mean width; close to the axis, the height of the central coffers; on the right side, the total width of the row; below the axis, the total height of the ceiling. (3) There is a lack of regularity in the foreshortening. The remotest horizontal line of the ceiling produces an interval as high as the previous one, probably because of the mix-up between this line and the one that marks the boundary of the coffered space. In perspective, however, two equal intervals ought to be of different heights. (4) The fresco shows system. In St Francis Preaching before Pope Honorius III, there are 3 out of 19 in the bracciosoldi system, and 7 out of 19 in the piede-palmi system. In the Recovery of the Wounded Man of Lerida, there are 8 out of 24 in the braccio-soldi system, and 8 out of 24 in the piede-palmi system. If we combine all the measurements, there are 13 out of 54 (24%) that are close to an integer value in the first system; and 18 out of 54 (33%) in the second one. The number of values being superior to 50, it is comparable to chance level. In the braccio-soldi system, integer values fall on every 29,95 mm and 49,92 mm ± 3 mm. One braccio (LCM of oncia and soldo) contains twelve oncie error intervals and twenty soldi error intervals, four of which are overlapping. The probability of obtaining an integer value ± 3 mm in the LCM interval is therefore equal to € 6(20+12-4 ) 599,00

= 168 599 = 28%. In the piede-palmi system, integer values fall on every 22,69 mm and 30,25 mm ± 3 mm. One palmo (LCM of pollice and dito) contains three pollici error intervals, four dita minuti error intervals, among which one in common. The probability of getting an integer value in the LCM interval is € 6(3+4 -1) 90,75 = 36 90,75 = 39%. Theoretical and empirical numbers of integer values are of the same order of magnitude: 24% ≈ 28% and 33% ≈ 39%.

Thereby, empirical concordances do not exceed random level. The hardly higher result in the piede-palmi system is due to the pollici and dita minuti narrower step. 4. Finally, suppose a variation of the units used by craftsmen around the ones historically documented. It is necessary to introduce such a variation, because the usual medieval units were not as accurate as present standards. This variation results from several factors: (1) Historical fluctuations. For example, fifteenth-century architectural measured surveys set the braccio fiorentino at 0,5875, 0,5860 or 0,5836 m26 ; (2) Professional fluctuations. Braccio and piede values were specific to different crafts, as proved by their names: "agrimensorio", "da legname", "d a muratori", "da panno", "da mercatori", etc.; (3) Regional fluctuations. In the sole case of Tuscany, the braccio took on the common value of 0,584 m (in Arezzo, Florence, S. Miniato, Pistoia, Siena, Montepulciano, Lucca, Pisa, Volterra, etc.), but shorter and longer values of the braccio were nevertheless used in Fivizanno (0,486 m), Massa (0,495 m), Montecarlo (0,593 m), and Pontremoli (0,692 m)27 ; The actual or putative presence of foreign masters (Cavallini, Rusuti, Giotto, etc.) on the site of the Upper Church precludes the exclusion of one or another of the units used in Assisi (br. 0,599 m; p. 0,363 m), Rome (br. 0,636 m; p. 0,298 m) or Florence (br. 0,584 m).

In order to take into account historical, professional and regional fluctuations, let us pro- to a standstill after the division by the smallest subunit. This conclusion exhibit a significant discrepancy with respect to the arithmetical hypothesis that requires discontinuity, not continuity.

ABSENCE OF SIMPLE PROPORTIONAL RATIOS

There is another possibility to save the arithmetical hypothesis. Just imagine that painters used arithmetical formulas with non-metric instruments as lines that can be folded in equal parts to determine a given ratio. In this case, arithmetic still works, though without standard dimensions. Suppose that, in the classical manner, duecento and trecento painters used modules without unit. Due to the use of arithmetical method, homologous parts should nevertheless be in simple proportional ratio a n /a n+1 . This is exactly what Andrés de Mesa supposes in taking a 1 /a 2 = 2 (op. cit., p. 34, fig. 8). But, in fact, such ratios are at random in the frescoes examined. In the Confirmation of the Franciscan Rule, the ratios It is also necessary to submit to analysis de Mesa's argument (op. cit., pp. 43-45) that painters used the superbipartiens rule in order to plan the foreshortening. This proportionality rule consists in making an interval two-thirds as high as the previous one. Alberti already criticized this method:

Hic essent nonnulli qui unam ab divisa aequedistantem lineam intra quadrangulum ducerent, spatiumque, quod inter utrasque lineas adsit, in tres partes dividerent. Tum huic secundae aequedistanti lineae aliam item aequedistantem hac lege adderent, ut spatium quod inter primam divisam et secundam aequedistantem lineam est, in tres partes divisum una parte sui excedat spatium id quod sit inter secundam et tertiam lineam, ac deinceps reliquas lineas adderent ut semper sequens inter lineas esset spatium ad antecedens, ut verbo mathematicorum loquar, superbipartiens […] 28 .

The passage can be interpreted in several ways. The narrowest interpretation consists in attaching more importance to the two-thirds rule than to the name of superbipartiens proportion that Alberti gave to it. The broader interpretation acts on the contrary: it considers the two-thirds rule as only an instance of the general case of superbipartiens proportion.

Works studied in this paper do not match the narrow interpretation. Take again The Recovery of the Wounded Man of Lerida that presents a 15 × 5 checkered ceiling. We can compare its intervals to two superbipartiens series, resulting from the foreshortening of the largest interval: The broad interpretation is based on the fact that the superbipartiens proportion concept regards a class of ratios. This concept comes from the medieval theory of proportions. 29 :1) , n, p, q natural integers, n = 1, p < q, p ≥ 2. Among the species of this genus, the superbipartiens 30 proportion meets the additional condition that p = 2. The two-thirds rule € (1+ 2 3 :1) is then given for q = 3. It is a subspecies of the superbipartiens species. It is thus possible to read in the De Pictura a foreshortening rule extended to any proportional ratio € (1+ 2 q :1), q ∈ N, q > 2 . Unfortunately, none of such ratios correspond to the intervals of the Recovery of the Wounded Man. The series are diverging as q is growing, so the best matching occurs for q = 3, which is an unsatisfactory case. 31 Consequently there is no proof that superbipartiens rule has been used.

€ 76 2 3 = 50,
We can nevertheless imagine a latissimo sensu interpretation of the rule, by extending it to all the cases in which a given term of the series is a constant ratio of the previous one. The rule is thus widened to multiplex, superparticularis, superpartiens, multiplex superparticularis and multiplex superpartiens proportions. This interpretation does not match any better The Recovery's intervals, because they do not follow a constant ratio. The series 0,827 (43/52); 0,867 (52/60); 0,909 (60/66); 0,868 (66/76) can be compared to the corresponding ratios of a checkered pattern in linear perspective. Establish first that constant ratios are antagonistic with perspective. 1) Lines AB, CD, EF, GH… are horizontal, and points A, C, E, G… collinear (Figure 5).

Therefore, ∠BAC = ∠DCE = ∠FEG… 2) By hypothesis, intervals are in constant ratio, hence AC / AB = CE / CD = EG / EF… also be found in less-known treatises, such as this anonym Tractatus proportionum: "Prima species superpartienti generis est proportio superbipartiens, que fit quando maior numerus continet totum minorem et insuper eius duas partes, ut 5 ad 3, 7 ad 5. Secunda species supradicti generis est proportio supertripartiens, que fit quando maior numerus continet in se totum minorem in se et insuper eius tres partes, ut sunt 7 ad 4, 11 ad 8, etc.", Saint-Dié, B. M., ms. 42, fol. 119r. 30 Supertripartiens, superquadripartiens, and superquinquepartiens proportions were formed on the same pattern.

3) After Elements, VI, 7, it follows that ∠ABC = ∠CDE = ∠EFG… 4) Since AB, CD, EF, GH… are parallels, the diagonals BC, DE, FG… are parallel to each other and cannot intersect whereas linear perspective requires a concurrent point (distance point).

Consequently, successive equal intervals in linear perspective cannot be in constant ratio. Reciprocally, a painting in which successive equal intervals are in constant ratio is not a linear perspective.

Knowing that the Recovery's intervals are not in constant ratio, its ceiling foreshortening ought to be compared to linear perspective. Empirical values are already known. Theoretical values can be found by means of analytic geometry. Begin with a simplified scheme of the fresco, on which the coordinates (x, y) of the points useful for obtaining theoretical values are marked (Figure 6). 31 To match the observed series to the superbipartiens series, a non integer q must be taken (optimal matching is for q = 2,3 that provides the terms The ceiling of the Recovery of the Wounded Man of Lerida uses a foreshortening method that is indiscernible from the one that is required by linear perspective. Consequently no superbipartiens method-understood stricto sensu, lato sensu or latissimo sensu-has been used in the laying out of the fresco. The fact opposes the arithmetical hypothesis. Why is it that painters of the Upper Church did not use proportionality ratios to solve the problem of the foreshortening? Perhaps because they took the representational problem as geometers, whose frame of mind always favors a continuum.

LENGTH OF THE OPERATING SERIES 32

The apparent simplicity of arithmetical method is partly due to the fact that it is deliberately vague about certain operations. For instance, the proportionality of intervals is established on parallels lines, but the existence of those parallels is taken for granted, whereas they need to be built. Many pre-perspective paintings such as The Approval of the Franciscan Rule, St Francis Preaching before Pope Honorius III, or Christ among the Doctors present an axis of symmetry. But to draw an axis of symmetry by means of proportional ratios is not an easy matter. It is thus necessary to describe all the constructive operations in order to reasonably compare geometrical and arithmetical methods in perspective.

Consider again The Recovery of the Wounded Man of Lerida, and, in order to neutralize all previous objections, suppose that intervals obey simple dimensions and proportions. Assume the layout to be a ruler-and-compass construction, according to the usual devices of geometry. The operating series are describable at different scales. One should distinguish between m.o. macrooperations (draw a perpendicular, divide a line into n equal parts…) and e.o. elementary operations (take a given aperture of a compass, join two points with a ruler…) For example, the m.o. "draw a perpendicular to a given line" contains five e.o.: "fix the metal point of the compass on a point of the given line", "draw a circle of any aperture", "fix the metal point on another point of the given line", "draw a circle of same aperture", "join the circles' intersections with a ruler" (Elements, I, 11, Figure 7).

Arithmetical method operating series (Figure 9)

1° Draw axis OS (take two marks, draw a vertical), 7 e.o.

2° Draw first horizontal

B 2 C 3 … B 1 C 2 D 3 … C 1 D 2 E 3 … on the
left side, and by diagonals

P 1 O 2 N 3 … O 1 N 2 M 3 … N 1 M 2 L 3 …

REINTERPRETING PERSPECTIVE ANOMALIES

The comparison of arithmetical and geometrical methods in early perspective shows the prominence of geometrical devices in duecento and trecento paintings. This is a somewhat counter-intuitive finding. One should then clarify the gap between the present result and the usual opinion on the topic, interpreting anew the main errors affecting medieval paintings.

1. The arbitrary behavior of small elements, such as the abaci of capitals, is not an exclusive clue of arithmetical method. Such anomalies are understandable considering the material constraints exerted upon the laying out. Fast-drying coating required a quick transfer of the pictorial scheme onto the intonaco. Consequently only the main lines were transferred, not the small elements.

2. The convergence of edges on a vanishing region is hardly a better proof in favor of an arithmetical formula. Methodology shows that all perspective painting-whether derived from an arithmetical or geometrical formula-is subject to errors of construction. In a geometrical layout, an error of parallelism is sufficient to make vanishing lines deviate.

3. The presence of a vanishing axis is not so easily attributable to errors of construction.

This pictorial scheme indeed presents a systematization that departs from random. The chances for architecture (edifichare) and perspective to cross-fertilize each other benefited from the high popularity of Vitruvius' De architectura in the Middle Ages. [START_REF]Le texte de Vitruve n'a cessé d'être connu (et donc recopié) de l'Antiquité à la Renaissance. Aussi ne faut-il pas s'étonner du nombre relativement important (près d'une centaine) de manuscrits aujourd'hui recensés qui contiennent des extraits, des parties ou l'ensemble du De architectura[END_REF] According to

Vitruvius, the architect ought to be "peritus graphidos, eruditus geometria [et optices non ignarus]" [START_REF]He must be well-read, expert in drawing, learned in geometry [and not ignorant in optics[END_REF] . He explicitly deals with perspective (scaenographia), referring to Agatharcus, Democritus, and Anaxagorus 40 . Except for Agatharcus who was working as a painter, his sources are in the field of optics and geometry. Vitruvius attributes a minor position to arithmetic: "Per arithmeticen uero sumptus aedificiorum consummantur, mensurarum rationes explicantur difficilesque symmetriarum quaestiones geometricis rationibus et methodis inueniuntur" [START_REF] Vitruvius | [END_REF] .

Medieval craftsmen seem to have appreciated in the same way the affinities between perspective, optics, and geometry. For instance, a Pistoia pulpit epigraph reports that Giovanni Pisano was learned in optics: "Sculpsit Johannes… doctum super omnia visa". Similarly, Villard de Honnecourt relies on geometry to introduce the art of drawing: "Ci comence li force des trais de portraiture si con li ars de iometrie les ensaigne" [START_REF] Hahnloser | Kritische Gesamtausgabe des Bauhüttenbuches[END_REF] . At folios 20r-21r, Villard subsequently presents some devices to measure an inaccessible height or distance, a usual problem of practical geometry and perspective.

EARLY PERSPECTIVE IN THE CLASSIFICATION OF SCIENCES

The medieval classification of sciences [START_REF] Weisheipl | Classification of the sciences in medieval thought[END_REF] can help us to appreciate the links between perspective, geometry and arithmetic. According to the classification by al-Fârâbî, transmitted to the Latin World by Gerard of Cremona's and Dominicus Gundissalinus' translations [START_REF] Roche | La classification des sciences de Gundissalinus et l'influence d'Avicenne[END_REF] , perspectiva is threefold: optica (direct rays), catoptrica (reflected rays), dioptrica (refracted rays). Picto-rial perspective is tied to optica only. The relations between sciences are described by the Aristotelian concept of subalternation. [START_REF]Aristotle rejects the genres mixing in the course of a demonstration but admits, under certain conditions[END_REF] There is subalternation when a superior science (scientia subalternans) provides the propter quid of a fact presented by an inferior science (scientia subalternata). Ever since Aristotle's Posterior Analytics, optics has been subordinated to geometry. This has given rise either to its outright absorption by geometry, as in Boethius' De Trinitate, or to its insertion in the geometrical sciences, as in Dominicus de Clivassio's Quaestiones perspectivae. [START_REF]Est sciendum quod quinque sunt scientiae mathematicae, scilicet arismetrica, geometria, musica, astrologia et perspectiva quae differunt secundum quod visum in prima conclusione; secundo sciendum quod ordo istius ad alias mathematicas est iste: quoniam arismetrica et geometria ordine doctrinae ista praecedunt, sed astrologia et musica ipsa secuntur[END_REF] Many classifications made a marked distinction between theoretical and practical sciences, as in the Etymologiae by Isidorus of Sevilla or the Didascalion by Hugh of St Victor.

In the Arabic tradition, on the contrary, scholars instead imagined a continuous gradation from the speculative to the practical sciences. [START_REF]The insertion of practical sciences in the classification of sciences seems to be of a Greek origin. Pappus reports that Heron's disciples divided mechanics into theoretical parts: geometry, arithmetics, astronomy, physics, and manual parts: architecture (oi) kodomikh/ ), ironworks (xalkeutikh/ ), carpentry (tektonikh/ ), and painting (zwgrafikh/ ), PAPPI ALEXANDRINI Collectionis quae supersunt[END_REF] Along the lines of Fârâbî, Dominicus Gundissalinus names seven mathematical sciences, having both theoretical and practical aspects, including optics (de aspectibus), statics (de ponderibus), and engineering (de ingeniis). Relying on the same tradition, Roger Bacon devoted an entire chapter of his Communia mathematica to Geometria speculativa et practica, [START_REF] Bacon | Communia mathematica Fratris Rogeri[END_REF] and fra' Luca Pacioli expounded the "parte principale de tutta l'opera de Geometria, in tutti li modi theorica e pratica." [START_REF] Pacioli | Summa de aritmetica, geometria, proportione et proportionalita[END_REF] Such connections explain why early perspective depended mainly from geometrical sciences and why, though a practical art, it was able to benefit from the contribution of speculative geometry and perspectiva naturalis.

Last, but not least, it must be mentionedthat mathematical sciences were dichotomized according to their subject: arithmetic as a science of discrete quantities (plh= qoj), geometry as a science of continuous quantities (me/ geqoj). This dichotomy was in place from the time of the works by Aristotle, Proclus, and Geminus, [START_REF] Vitrac In | [END_REF]

Figure 2 :

 2 Figure 2: The Confirmation of the Franciscan Rule

Figure 3 :

 3 Figure 3: St Francis Preaching before Pope Honorius III

Figure 4 :

 4 Figure 4: The Recovery of the Wounded Man of Lerida

  duce a continuous variation of braccio from 525,6 mm (br. fl. -10%) to 699,6 mm (br. rom. + 10%) and then study the fluctuation of the number of integer values y as a function of this extensible braccio x. If function y = f(x) admits a maximum almost equal to the total number of the frescoes' measurements, this maximum will be the value of the unit searched for. An optimization algorithm enables us to detect the maxima. The function admits 10 values (550,01 ≤ br. ≤ 550,24) as a minimum minimorum, and 27 values (652,40 ≤ br. ≤ 652,44) as a maximum maximorum. The range of integer values plus or minus the margin of error is from € 10 54 = 18% to € 27 54 = 50%, i.e. less than a half. Consider now a continuous variation of the piede from 268,2 mm (p. rom. -10%) to 399,3 mm (p. umbro + 10%). The number of integer values fluctuates from 12 (375,34 ≤ p. ≤ 375,54) to 31 (273,30 ≤ p. ≤ 273,35). Thus, the range of integer values plus or minus the margin of error is from € 12 54 = 22% to € 31 54 = 57%. It follows that none of the unit systems can convert the dimensions measured on the frescoes into integer or simple measures. The frescoes' dimensions reveal no numerical consistency and, in nearly half of the cases, they come

  75. The result is even clearer in the third fresco. The Recovery of the Wounded Man of Lerida allows us to calculate similar ratios from more accurate mean values. But ratios € 79,00 85,03 = 0,9290…, € 85,03 91,99 = 0,9243…, € 91,99 100,09 = 0,9190…, € 100,09 108,84 = 0,9196…, and € 108,84 118,23 = 0,9205… fluctuate around € 12 13 = 0,9230…, which is an unlikely fraction, owing to its denominator. Therefore, craftsmen seem to have left aside elementary fractions.

  Figure 5

Figure 6 .

 6 Figure 6. Scheme for the Recovery of the Wounded Man of Lerida

Figure 7

 7 Figure 7

  on the right side. These diagonals converging on lateral points T T´ are exactly those that should be used to obtain the perspective foreshortening. If medieval painters had ever followed the arithmetical method, why are the horizon and diagonals lines so neatly visible? This correspondence is not occasional. A squared coffer divided by the diagonal is usual in Assisiate frescoes. It works as a spatial roofing pattern in several scenes: St Francis Honored by a Simple Man, St Francis before the Sultan (Trial by Fire), Death of the Knight of Celano, and Christ among the Doctors. Thus we cannot exclude that such diagonals could have served as an empirical rule to place receding lines in depth.

  15 G.J. KERN, Die Anfänge der zentralperspektivischen Konstruktion in der italianischen Malerei des 14. Jahrhunderts , «Mitteilungen des Kunsthistorischen Instituts in Florenz», 2, 1913, pp.39-65; E. PANOFSKY, Die Perspective als symbolische Form, «Vorträge der Bibliothek Warburg», 1924, pp. 258-331, French transl. La Perspective comme 'forme symbolique', Paris, Éditions de Minuit, 1975; D. GIOSEFFI, Perspectiva artificialis. Per la storia della prospettiva, spigolature e appunti, Trieste, Tipografia Smolars, 1957. The thesis has been amended by R. SINISGALLI, Per la storia della prospettiva, 1405-1605, Roma, L'Erma, 1978.

	O
			(1)	DE EF	=	AB BC	(2)	OE ED	=	OB BA	(3)	OE EF	=	OB BC
	D	E	F
	A	B €	C
	Figure 1: The arithmetical hypothesis

  22,69 mm)22 .

			br.	o.	s.	q.			p.	pa.	po.	d.
		br.	1		12	20	60		p.	1	4	12	16
		o.	1 12		1	5 3	5		pa.	1 4	1	3	4
		s.	1 20		3 5	1	3		po.	1 12	1 3	1	4 3
	€	q.	1 60	€	1 5	1 3	1	€	d.	1 16	1 4	3 4	1
	€	€ braccio umbro (0,599 m)	€	€ piede umbro (0,363 m) €
	€	€		€				€	€	€
	19 A similar case occurs in Lescot's façade for the Louvre, whose measurements are multiple of the pied du Roi
	(326,6 mm) used in Paris around 1546, J.-P. SAINT AUBIN, Photogrammétrie et étude des ordres: le Louvre de
	Lescot, in L'Emploi des ordres à la Renaissance, ed. by J. Guillaume, Actes du colloque de Tours (9-14 juin
	1986), Paris, Picard, 1992, pp. 219-226.
	20 K. HECHT, Maßverhältnisse und Maße der Cappella Pazzi, «Architectura», 6, 1976, pp. 148-174.
	21 J. GUILLAUME, Désaccord parfait: ordres et mesures dans la chapelle des Pazzi, «Annali di Architettura», 2,
	1991, pp. 9-23.

22

"Brachium continet 12 vntias", "Pes palmorum quattuor, pollicum seu vnciarum duodecim, digitorum vero sexdecim". We leave aside the quattrino, whose narrow step (9,98 mm) is not differential enough. R.E. ZUPKO, Italian Weights and Measures from the Middle Ages to the Nineteenth Century, Philadelphia, The American Philosophical Society, 1981, pp.

47-48 (braccio), 197 (piede)

.

Look now at The Confirmation of the Franciscan Rule, which de Mesa (op. cit., fig.

11-

  A 1 P 1 ⊥ OS (draw a perpendicular), 5 e.o. 3° Draw horizontal A 2 P 2 at the distance A 1 P 1 from H 1 H 2 (draw a parallel), 6 e.o. 4° Calculate H 2 H 3 = k. H 1 H 2 (apply a proportional ratio), 2 e.o. 5° Draw horizontal A 3 P 3 at the distance A 2 P 2 from H 2 H 3 , 6 e.o. Repeat 3 times operations 4 and 5 to get lines A 4 P 4 … A 6 P 6 , 24 e.o. I 1 (apply a ratio), 3 e.o. 13° Draw H 1 I 1 on A 1 P 1 on both sides of axis OS (draw a circle of radius I 6 on A 6 P 6 on both sides of axis OS, 2 e.o. 18° Transfer interval H 6 I 6 little by little onto A 6 P 6 with a compass, 15 e.o. 19° Join A 1 A 6 with a ruler, 1 e.o. Repeat 15 times operation 19 to get all segments B 1 B 6 … P 1 P 6 , 15 e.o. I 1 on A 1 P 1 on both sides of axis OS (draw a circle of radius I 1 little by little onto A 1 P 1 with a compass, 15 e.o. 6° Draw pencil lines A 1 O … P 1 O with a ruler, 16 e.o. This difference in length is not due to 33 The most popular text was Adelard of Bath's version of the Elements: EUCLID OF ALEXANDRIA, The First Latin Translation of Euclid's Elements commonly ascribed to Adelard of Bath: Books I-VIII and Books X.36-XV.2 ed. by H.L.L. Busard. Toronto, The Pontifical Institute of Medieval Studies, 1983 (for a mathematical commentary, see EUCLID, Les Éléments, ed. by B. Vitrac, 4 vols, Paris, PUF, 1990-2001), and its commentary by AL-NAYRÎZÎ, Anaritii in decem libros priores Elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis, edidit M. Curze, Leipzig, B. G. Teubner, 1899. Leonardi Pisani Practica geometriae ed opuscoli, Roma, Tipografia delle scienze matematiche e fisiche, 1862); ABÛ AL-WAFÂ' AL-BÛZJÂNÎ, Kitâb fî mâ ya h tâju ilayhi al-s âni' min a'mâl al-handasa / L'oeuvre scientifique d'Abû al-Wafâ' al-Bûzjânî, II. La géométrie pratique d'Abû al-Wafâ', ed. J. Aghayani Chavosi, thèse de doctorat, Université de Paris VII, 1997; N. CHUQUET, La Géométrie. Première géométrie algébrique en langue française (1484), introduction, texte et notes par Hervé L'Huillier, Paris, Vrin, 1979). For further references, see the works quoted at note 36.arithmetical method complexity, but to the fact it presupposes a great number of background operations. Once the implicit operations have been rendered, the method is much more tedious.5. COINCIDENCE OF POINTS AT INFINITY WITH VISIBLE LOCISince points at infinity (central and lateral vanishing points) are useless according to arithmetical method, painters should not have used them. Consequently, they should not coincide with visible loci of the pictorial composition. In contrast, if a systematic and accurate correspondence is found, it will support the idea that duecento and trecento painters actually used geometrical devices instead of arithmetical ones.The systematic correspondence with visible loci of the composition is already a feature of certain paintings of the Giotto epoch. For instance, in The Recovery of the Wounded Man, central point O and lateral points T T´ are aligned, thus forming a line: the so-called horizon of classical perspective. But with the side ceiling being divided in six by four coffers, it appears that this line coincides with a perfectly visible line that divides lateral ceilings in two equal parts. In addition, the checkered ceiling is marked by diagonals A 1

		12° Calculate € 14° Transfer interval H 1 I 1 little by little onto A 1 P 1 with a compass, 15 e.o. 1 1 2 H 1 I 1 ), 2 e.o. 15° Calculate K = k 5 , 5 e.o. A1 B1 C1 D1 E1 F1 G1 H1 I1 J1 S K1 L1 A2 A3 A4 A5 2 H 1 € 16° Calculate 1 2 H 6 I 6 = K. 1 2 H 1 I 1 (apply a ratio), 3 e.o. A6	M1	P6	N1 P5	P4	O1 P3	P2	P1
		€ 17° Draw H 6 m.m.o. = 34 €	m.e.o. = 111
								S
	A1 T	B1	C1	D1	E1	F1	G1	H1	I1	O	J1	K1	L1	M1		N1		O1	P1
		Figure 10. Geometric layout for the Recovery of the Wounded Man
	A2																	P2
		A3																P3
		A4															P4
			A5													P5
			A6												P6
								O
		Figure 9. Arithmetical layout for the Recovery of the Wounded Man
		Geometrical method operating series (Figure 10)
		3° Calculate	1 2 H 1 I 1 (apply a ratio), 3 e.o.
		€ 4° Draw H 1 € 5° Transfer interval H 1 7° Draw horizontal OT ⊥ OS (draw a perpendicular), 5 e.o.			1 2 H 1 I 1 ), 2 e.o.
	8° Draw diagonal H 1 T, 3 e.o. 34 See the treatises by BOETHIUS (M. FOLKERTS, «Boetius» Geometrie II: Ein mathematisches Lehrbuch des
	Mittelalters, Wiesbaden, Steiner Verlag, 1970); ABRAHAM BAR H IYYA (M. CURZE, Der Liber Embadorum des 9° Draw horizontal A 2 P 2 through point H 1 T ∩ G 1 O …, 6 e.o. Abraham bar Chijja Savasorda in der Übersetzung des Plato von Tivoli, Leipzig, B. G. Teubner, 1902); Artis
	Repeat 4 times operation 9 to get all lines A 3 P 3 … A 6 P 6 , 24 e.o. cuiuslibet consummatio (S.K. VICTOR, Practical Geometry in the High Middle Ages, Philadelphia, The Ameri-
	can Philosophical Society, 1979); Leonardo FIBONACCI (Scritti di Leonardo Pisani, matematico del secolo deci-
	m.m.o. = 13 moterzo, publicati da B. Boncompagni, vol. II. m.e.o. = 83

1° Draw axis OS (draw a vertical), 3 e.o. 2° Draw first horizontal A 1 P 1 ⊥ OS (draw a perpendicular), 5 e.o. The comparison of the operating series is definitely unfavorable to arithmetical method because it requires more operations (34 m.m.o. and 111 m.e.o.) than does geometrical method (13 m.m.o. and 83 m.e.o.). Furthermore, one should note that the present comparison has been limited to the devices of learned geometry, 33 supposing all constructions to be drawn with ruler and compass. But practical geometry 34 freed oneself from such constraints, by having recourse to many geometrical instruments. The fact has been known since earliest antiquity. Introduce only the main instruments documented in the thirteenth century: ruler, compass, square, level, and libella. Thus, arithmetical method appears to be twice as long (34 m.m.o. and 111 m.e.o.) as the geometrical-instrumental one (13 m.m.o. and 63 m.e.o.).

  The use of axial composition could have resulted from the theory of binocular vision expounded by Ibn al-Haytham, latinized as Alhazen (Kitâb al-manâzir / De aspectibus, III, 2, 12 sq.), and by his Latin commentators: Roger Bacon, Perspectiva, II, II, 2; John Pecham, Perspectiva communis, I, 80; and Witelo, Perspectiva, III, 27 sq. This idea has been tested onto works whose edges converge on two central vanishing points. 35 They correspond to the case of homonym diplopia discerned by Ibn al-Haytham.

	«Arabic
	Sciences and Philosophy», 13, 2003, pp. 79-99; ID., Une application méconnue des principes de la vision bin-
	oculaire: Ibn al-Haytham et les peintres du Trecento (1295-1450), «Oriens/Occidens», 5, 2004, pp. 93-131.
	36 On practical geometry, see A. SIMI and L. TOTI-RIGATELLI, Some 14th and 15th century texts on practical
	geometry, in M. FOLKERTS and J. P. HOGENDIJK, eds., Vestigia Mathematica. Studies in Medieval and Early
	Modern Mathematics in Honour of H.L.L. Busard, Amsterdam/Atlanta, Rodopi, 1993, pp. 453-470; A. SIMI,
	Problemi caratteristici della geometria pratica nei secoli XIV-XVI, Scienze mathematiche e insegnamento in
	epoca medioevale, a cura di P. Freguglia, L. Pellegrini e R. Paciocco, Napoli, ESI, 2000, pp. 153-200.

7. EARLY PERSPECTIVE IN THE EDUCATIONAL CONTEXT

Inserting arithmetical and geometrical methods in the educational context is suitable for testing craftsmen's possibilities of applying mathematics in workshop practices. Perspective as a science took its departure with Piero della Francesca's first geometrical proofs. Before this date, there was-strictly speaking-no more than affinity between perspective and the medieval sciences.

1. The common opinion is that painters and craftsmen, not being admitted to the university, benefited only from abacus school training. This could be a sound argument for the arithmetical hypothesis-as abacus means properly arithmetic-but any confusion between abaco and scuola d'abaco should be avoided. Arithmetic was not the one and only teaching of abacus schools. Arismetricha, geometria, 36 edifichare e prospettiva 37 were also on the program, as at Santa Trinità in Florence, founded by Paolo Dagomari dell'Abaco. Thus, there are some reasons to think that it was there that medieval painters acquired basic notions of geometry and optics they afterwards put into practice.

35 D. RAYNAUD, Ibn al-Haytham sur la vision binoculaire: un précurseur de l'optique physiologique,

37

On practical perspective, see G. ARRIGHI, Un estratto dal "De visu" di M° Grazia de'Castellani (dal Codice Ottoboniano latino 3307 della Biblioteca Apostolica Vaticana), «Atti della Fondazione Giorgio Ronchi», 22, 1967, pp. 44-58; F. CAMEROTA, Misurare "per perspectiva" : geometria practica e prospectiva pingendi, in La prospettiva, a cura di R. Sinisgalli, Firenze, Cadmo, 1998, pp. 340-378; F. CECCHINI, Ambiti di diffusione del sapere ottico nel Duecento. Tracce per uno studio sulle conoscenze scientifiche degli artisti italiani del XIII secolo, in L'Artiste et l'Oeuvre à l'épreuve de la perspective, op. cit., pp. 19-42.

  right up to the Italian Renaissance treatises that still identified devices per numero and per linea.51 Having recourse to these categories, we understand better why the drawing of the frescoes in the Upper Church at Assisi presents such poor affinities with arithmetical method. Though further research is needed on medieval paintings, it appears that continuous geometrical insight could have served as a guide to the earliest perspec-

	tive experiments.

51 L. PACIOLI, Divina proportione, Venice, Paganino de Paganini, 1509; P. CATANEO, L'Architettura, Venice, Aldo, 1567; A. PALLADIO, I Quattro Libri de architettura, Venice, Domenico dei Franceschi, 1570. About the devices per numero and per linea, see S. GESSNER, Les Mathématiques dans les écrits d'architecture italiens, 1545-1570, thèse, Paris, Université de Paris VII, 2006, pp. 109-144.

E. PANOFSKY, Die Perspective als symbolische Form, op. cit., p. 125.

A. DE MESA GISBERT, El 'fantasma' del punto de fuga en los estudios sobre la sistematización geometrica de la pintura del siglo XIV, «D'Art», 15, 1989, pp. 29-50. The author has henceforth specialized in architectural measured survey.

Consequently, arithmetical method supports the idea that perspective was a Renaissance invention: the author speaks of Brunelleschi's contribution, Ibid, p. 35. For criticism, see D. RAYNAUD, L'Hypothèse d'Oxford, Essai sur les origines de la perspective, Paris, PUF,

1998, pp. 4-9, 132-150.

We have relied upon the photographic survey by B. ZANARDI, C. FRUGONI, F. ZERI, Il Cantiere di Giotto, p. 128. The survey scale can be deduced from the dimensions of the fresco (363 × 357 cm), the height of the standing St Francis (122 cm), the height of the figures (122, 111, 123 cm), and the diameter of the halo (35,5 cm). We have systematically checked the parallelism and lack of distorsion on the fresco, following D. RAYNAUD, La théorie des erreurs et son application à la reconstruction des tracés perspectifs, in L'Artiste et l'Oeuvre à l'épreuve de la perspective, ed. by M. Dalai Emiliani, M. Le Blanc, P. Dubourg Glatigny, Rome, École Française de Rome, 2006, pp. 411-430.

B. ZANARDI, Il Cantiere di Giotto, op. cit., p. 242.

These braccio values are given by K. HECHT, Maßverhältnisse und Maße der Cappella Pazzi, op. cit.; L. BENEVOLO, S. CHIEFFI, G. MEZETTI, Indagine sul S. Spirito di Brunelleschi, «Istituto di Storia dell'Architettura, Quaderni», 85/90, 1968, pp. 1-52; C.L. FROMMEL, Der Römische Palastbau der Hochrenaissance, 3 Bde, Tübingen, Verlag E. Wasmuth, 1973.

 27 R.E. ZUPKO, Italian Weights and Measures, op. cit., p. 46.

For details, see D. RAYNAUD, Las primeras perspectivas de los siglos XIII y XIV según el enfoque del modus operandi, in Perspectiva: fundamentação teórica e cultural, ed. by Magno Mello, Belo Horizonte, Argumentum, 2009, pp. 41-62.


Why did Geometrical Optics not lead to Perspective in Medieval Islam?

Rationality and Good Reasons in the Anthropology of Mathematics Dominique Raynaud * In the foreword to his Studies in the Sociology of Religion, Max Weber questions the conditions under which many cultural phenomena appeared as Western singularities, but nonetheless took a universal value (Weber, 1922). He wonders in particular about the rise of linear perspective in the West. His general solution to the problem is based on the idea of a 'rationalization process' that grows up in determinate societies, and gradually pervades all human activities, according to the particular conditions men meet or create. Taking the perspective problem anew, it has been shown that the favourable conditions reputed to be specific to Renaissance go back in fact to the Medieval optical literature that played a prominent role in the rationalization of sight [START_REF] Raynaud | L'Hypothèse d'Oxford. Essai sur les origines de la perspective[END_REF]. There are many clues of this contribution, such as the borrowings and explicit references that are to be found in the treatises by Alberti, Piero della Francesca, Ghiberti, or Vinci, to begin with. But, on the other hand, a growing number of critical editions of Arabic optical literature now enables us to see that the Latin optical treatises, the Renaissance scholars relied on, comment thoroughly Arabic optics. As a result, we must recognize a second-order continuity between Renaissance and Medieval Arabic achievements that put forward new and specific questions.

The problem I want to put into consideration is the following: if Renaissance perspective proceeds from Medieval Latin optics, whose contents go back to Arabic optics, we could then expect the first stages of perspective advancement to be found in the Arabic civilization. But there is nothing to confirm this deduction, because in all the areas where Arabic was spoken-so including Persia, Central Asia, or al-'Andalûs-painting, miniature, and technical diagrams were drawn according to other types of representation: frontal and oblique projections, for the most part. So how to explain that first rank optical researches grew up in Islam without any application to representation? Is it the consequence of a decisive improvement of optical knowledge after been transplanted into Latin World? Is it due to the sparse diffusion of optical treatises in Orient? Is it a consequence of the well known Islamic aversion for images? The fact is unclear, and requires a new sight. After having detailed the nature and soundness of optical knowledge tied to perspective questions (Part I), I will scrutinize three sets of reasons: sociability, religious beliefs, and axiological orientation of those who were the potential users of perspective in Islam (Part II).

Part I. Arabic Theoretical Knowledge on Perspective

This section is devoted to establish that all optical sources that were actually combined to give birth to linear perspective were yet available in Medieval Islam.

Before establishing the point, I will say few words of the four main texts that will intervene in the course of the demonstration: (1) Euclid's Optics was translated into Arabic by Hiliyâ b. Sarjûn-a scholar of Greek origin as his name tell us: 'Elias filius Sergii'. His Kitâb Uqlîdis fî ikhtilâf al-manâzir (Book of Euclid on the Difference of Perspectives) dates from ca. 212 H. / 827-28 [START_REF] Kheirandish | The Arabic Version of Euclid's Optics (Kitâb Uqlîdis fî ikhtilâf al-manâzir)[END_REF] Tabaqât, the treatise was composed between 419-29 H. / 1028419-29 H. / -38 (Sabra, 1989)).

The Perspective of the Circle

Ibn Sarjûn: "Wheels of chariots are seen at times as circular and at other times as distorted (*). Let then be a wheel, ABGD, and we draw in it two diameters, AB, GD, intersecting at right angles at point E; and let the eye be on a plane parallel to the plane of the circle. Then if the line drawn from the eye's position to the circle's centre is perpendicular to the circle's surface, or else equal to half its diameter but not perpendicular, then the diameters are seen equal; therefore it is demonstrated that the wheel is seen as circular; but if the line drawn from the eye's position to the circle's centre is not perpendicular to the circle's surface and not equal to half its diameter, then the diameters are seen as unequal; therefore it is demonstrated that the wheel is seen in this case as distorted (*).

And that is what we wished to demonstrate" (Fig. 1 presupposes an affine transformation K = fa / fb = gc / gd (Fig. 2). This geometrical property was known from Archimedes ' Conoids and Spheroids, prop. 4 (1970: 166-170) and Ibrâhîm b.

Sinân expound it his Maqâla fî rasm al-qutû' al-thalâtha (Epistle on the Drawing of the Three Sections) (Rashed and Bellosta, 2000: 263-89). ** Al-Kindî's Rectification usefully complements the proposition, insisting the on continuity in experiencing the various perspectives of a circular figure. The point is made clear, in considering the rectilinear movement of a chariot passing along the spectator. Renaissance scholars will gradually put in evidence that a circle is always seen as a conic section-generally an ellipse-and that it can be obtained by an affine projection (Vinci, Codex Atlanticus, fol. 115rb, ca. 1510). This fundamental law of perspective was yet a source of geometrical interest, before the X th century.

Intersecting the Visual Pyramid

Ibn Sarjûn: "[Of] planes below the eye, the most distant one is seen at the highest. For let the eye's position be point A, which is higher than plane BDEF, and let the ray[s] AB, AD, AE, AF fall.

Then I say that FE is seen higher than ED, and ED [is seen] higher than DC. has point D on it, and since FE is seen because of GH, and ED is seen because of HI, then FE is seen higher than ED, and in the same way, ED is also seen higher than DC. And that is what we wished to demonstrate" (Fig. 3).

3 A B C D E F G H I Fig. 3
Al-Kindî: "Among surfaces disposed below the eye in the same plane, the most distant from the eye are seen higher than the nearer from eye […] Demonstration: Draw a line from A to B, and lines AF, AE, AD, AC, that we suppose to be the [visual] rays. The size they enclose is what the ends of the ray encompass. The ends of the ray under which we see EF are the two lines AF and AE, the ends of the ray under which we see DE are the two lines AD and AC and the ends of the ray under which we see CD are the two lines AC and AB. Draw a perpendicular at point C towards point G on AF, and mark point H where line GC cut line AE, and mark point I where line GC cut line AD (*). Then point C is seen by line AC and point D is seen by line AID; from A we see point H higher than point I according to the size of line IH. But point E is seen from A by line AHE, therefore point E is seen more deviated upwards, depending on the size of line IH (**). But point F is seen at the same time that point G, therefore ray AGF is higher than H depending on the size of line GH; therefore point F is seen higher, i.e. more deviated in the direction of A than point H, depending on the size of line HG (**). Therefore it is seen higher than E by line HG. As a result F is seen more deviated upwards than E, and E more deviated than D, and D more than C, and C more than B, by the same way. Therefore most distant places and surfaces, having all their width on line BF, are seen higher, i.e. more deviated in the direction of A, than the points of line BF nearest to A. And that is what we intended to demonstrate" (Fig. 3). 4 Commentaries. * Ibn Sarjûn and al-Kindî use explicitly the system known as the 'intersection of the pyramid of visual rays' in which picture plane GC intersect visual rays AF, AE, AD, AC… This is nothing but the definition of linear perspective given by Alberti (1992: 103), Vinci (1970: 34), or Danti (2003: 119, def. I). **, *** Al-Kindî goes beyond the Arabic version of the Optics for where Ibn Sarjûn considers only the position of segments GH, HI, IC (higher or lower), al-Kindî estimates the length of segments ("point H is seen higher… depending on the size of line IH"). So al-Kindî was perfectly aware that GH < HI < IC. But being line GH the picture plane, and expression as "size of line IH", "size of line HG" referring to the size of visible things, we must conclude that, from the IX th century, Arabic scholars knew how to size the diminution of the receding intervals, bringing forward the material required by linear perspective (Alberti, 1992: 121;Francesca, 1984: 85, prop. XXII, Danti, 2003: 188, th. XXIII). Even if Arabic scholars did not think about the practical consequences of this result, it was yet ascertained by way of demonstration.

The Route to the Vanishing Point

Al-Kindî: "The intervals between two parallel lines are seen unequal; what is remoter from the pupil is seen narrower, and what is closer to the pupil is seen larger (*). Suppose AC and BD to be the two parallel lines; suppose C to be near D, and let be E the position of the pupil […] Let be the intervals AB, FG, HI… Demonstration: AB is seen from E under angle AEB, the largest one; FG is seen from E under angle FEG which is smaller than angle AEB; and HI is seen from E under angle HEI which is smaller than angle FEG. Therefore AB is seen larger than FG and FG is seen larger than HI, for what is seen under a large angle is seen larger, and what is seen under a small angle is seen smaller. But the ends of the largest interval are seen more distant than the ends of the smaller one. Therefore the two points A and B, that are closer to the pupil, are seen more distant than the two points F and G, that are remoter from the pupil. By the same way, we demonstrate that the two points F and G are seen more distant than the two points H and I […] Therefore if the pupil is set on point E, the two points A and B are seen more distant than the two points K and L (*). And that is what we intended to demonstrate" (Fig. 4). 5

E A F H K C B G I L D Fig. 4
Qustâ b. Lûqâ: "By which reason the same object appear to be of an unequal size, when getting closer or moving away from the eye? It will be seen larger when getting closer, smaller when moving away […] Let be line AB the visible object, point E the seeing eye, and let be AEB the cone of rays spreading from the eye. Move the visible object AB from position AB to FG (**), then the ray that spreads from the eye, which is at point E, up to this object, is cone FEG; therefore the visible object is seen as FG under angle FEG. But angle FEG is smaller than angle AEB, therefore FG is seen smaller than AB. Move now the visible object from FG to HI (**), it will be seen under angle HEI, which is smaller than FEG. Imagine the visible object to be remove from HI to KL (**), it will be seen under angle LEK, which is smaller than HEI. Therefore KL is seen smaller than HI.

Similarly, if we move the object from KL to CD (**), then it will be seen under cone CED, and so will appear smaller than KL. Similarly, as the visible object moves away from the eye, the angle under which is seen is smaller and smaller. If the narrowing goes up to the limit that the angle is a line without breadth, the visible object which is the base of this angle will not be seen, due to its narrowing. The visible object goes on diminishing, according to the diminution of the angle, till becoming so small in size that the visual sense cannot perceives it no more (***)" (Fig. 4). 6

Commentaries.

Experiments with the Camera Obscura

Al-Kindî: "This [the rectilinear propagation of rays] will be still more evident and clear if we take a board and make a hole perpendicularly and regularly in it with a saw (*); if we then place the centre of the aperture made with a saw and the centre of a candle opposite each other, so that the line drawn from the candle cut orthogonally the candle's diameter and the aperture made with a saw; and finally if we place behind the opened board another board, the surface of which being opposite and parallel to the first board… For instance, let be circle ABG the candle, board DE the perforated board, and space UZ the aperture. And again, let be HT the board on which the light is falling, and let be space KL the part of the board the light is falling on. If therefore we draw a line from point K to point U, and extend it by a straight line, it will fall on point G of the candle, which is on the side opposite to Z (**)… Definitely, that would not happen if the limits of rays did not follow straight lines (***)" (Fig. 5).

7 A B G U H L T Z E D K Fig. 5
Ibn al-Haytham: "We also find that the radiation of all lights takes place only in straight lines and that no light radiates from a luminous object except in straight lines (***)-provided that the air or transparent body between the luminous object and the body on which the light appears is continuous and of similar transparency.

When this state of affairs is examined at all times it is found to be uniform, suffering no variation or change…

As for the light of the sun, when it [any light] enters through a hole into a dark chamber (*) … the light will appear to extend rectilinearly from the hole through which the light enters… And similarly with the light of the stars: for, in a moonless night, let any of the large stars (such as Venus, or Jupiter at its nearer position [to the earth], or also Mars at its nearer position, or Sirius) be opposite a hole giving into a dark chamber: its light will appear in the chamber and will be found opposite to the hole… If he [the experimenter] observes the star for some time until it has moved through an appreciable distance, its light in the chamber will be found to have moved from its [former] place so as to be rectilinearly opposite the star (**). And as the star moves, that light will move, and the light and the hole and the star will always be found to lie on a straight line" (Fig. 5). 8 Vinci, for instance, presents a very similar figure to Ibn al-Haytham's one, and emphasizes the same argument (Vinci, 1970: 45).

To sum up, all the scientific materials required for the setting up of perspective were yet in position, at least since the XI th century. Why then did linear perspective not arise in Medieval Islam?

Part II. Arabic Disregard for Practical Perspective

Although geometry was used for artistic purposes (Özdural, 2000) Museum, Rogers Funds). So, in all representational domains, if craftsmen ever used a compelling system, they chose prevailingly oblique or frontal projection. Three factors-not of equal importance-seem to have determined this choice.

The Sociability Factor: Artisans and Scientists

The relationship between artisans and scientists is poorly documented in Medieval Islam. Saliba (1999) and Özdural (2000) have nevertheless highlighted many points of interest from the mathematical literature itself. The main references to the topics come from two works of Abû al-Wafâ' al-Bûzjânî (d. 940): the first work is an arithmetic textbook addressed to the bureaucrats and state functionaries [START_REF] Medovoi | Ob arifmeticheskom traktate Abu-l-Vafy[END_REF]; the second one is a treatise on practical geometry intended for the artisans (Aghayani, 1997). There is also a testimony by Ibrâhîm b. Sinân (908-946) that his treatise on gnomons passed in the hands of an artisan he become acquainted with [START_REF] Saliba | Risâlat Ibrâhîm b. Sinân b. Thâbit b. Qurrâ fî al-ma'ânî al-latî istakhrajahâ fî alhandasa wa al-nujûm[END_REF]. Two other references are to be found in the treatises by Abû al-Rayhân al-Bîrûnî (d. ca. 1048): in the first tract, he informs us some makers of astrolabe of his time used precise arithmetic methods, instead of rough craft methods [START_REF] Saidan | Kitâb tastîh al-suwar wa-tabtîh al-kuwar[END_REF]; in the second tract, he describes exact and approximate methods for the calculation of the qibla, being the approximate ones designed for the architects and artisans (al-Bîrûnî, 1967). To these sources, we must add some other treatises. The first one is the

Kitâb fî al-jabr wa al-muqâbala (Book of [Calculus by] Restoration and Reduction) by

Abû Kâmil (d. ca. 930). In the second part of the book, Abû Kâmil embarks on the study of pentagon and decagon, two figures of practical interest for the design of geometrical patterns [START_REF] Yadegari | Abu Kamil's on Pentagon and Decagon[END_REF]. Another valuable treatise is the Miftâh al-Hisâb (Key of Arithmetic) composed by al-Kâshî (d. 1429), where some parts of Book IV, Chapter 9 are devoted to the calculation of qubbas and muqarnas [START_REF] Dold-Samplonius | The Volume of Domes in Arabic Mathematics[END_REF][START_REF] Dold-Samplonius | How al-Kâshî Measures the Muqarnas: A Second Look[END_REF].

Nevertheless, the most distinct evidence that mathematicians and craftsmen were joined together appears in two other documents. At the end of an untitled treatise on a geometrical problem involving the solution of a cubic equation, Omar Khayyam (d. 1131) tells us that the incentive to write it came from a question asked to him during a meeting with artisans-possibly in Isfahan (Amir-Moéz, 1963;[START_REF] Özdural | Omar Khayyam, Mathematicians, and 'Conversazioni' with Artisans[END_REF]. Two centuries ago, Abû al-

Wafâ' wrote his Kitâb fî mâ yahtâju ilayhî al-sâni' min a'mâl al-handasa (Book on What is

Required by Artisans of Geometric Constructions). In Chapter 10, he declares he "attended a meeting to which took part craftsmen and geometers that were asked about the construction of a square made of three squares." Such testimonies establish the existence of long standing contacts between the mathematicians and the lower class of craftsmen. And this makes most vivid the paradox that geometrical optics did not lead to linear perspective in Medieval Islam.

But we have to keep in mind that the relationship between artisans and scientists differed greatly in nature, according to the craftsmen the mathematicians addressed to.

Astrolabe and scientific instruments makers were generally praised for their ability. "All that is used by craftsmen is done without mathematical principle, and therefore cannot be trusted, and causes many errors… There is no doubt that, among all the constructions traced by artisans, the correct ones were borrowed from geometry. Indeed these constructions were imagined and demonstrated by the geometer, and then, by observation, the artisan took them in a superficial way, without thinking about the demonstration of their correctness. That is why artisans indulge so many errors" (Aghayani, 1997: 106, 119).

Similar statements can be found as regards the inscription of regular polygons in a circle, the construction of a square compound of a (non square) number of small squares, etc.

In sum the data are ambiguous, to say the least. On the one hand, the connexion between craftsmen and mathematicians seems to have been a widespread practise in Medieval Islam.

And such meetings were probably held to favour the magnificence of the arts, especially under the Buyid (934-1055) and Saljûqid (1037-1194) patronages. On the other hand, this connexion probably gave the best results in instrumental making than in decorative crafts, as mathematicians repeatedly complain about the crude and inaccurate methods of masons and mosaic makers. Had the mathematicians have thought to apply geometrical optics to art, they should not have carried out this task successfully, because of the bad reception of theoretical geometry by craftsmen. This is a first level of explanation of why linear perspective did not arise in Medieval Islam. But this sociological explanation, based on the spreading-and-closing of social networks, is insufficient.

The Religious Factor: Presumed Aniconism

Among the arguments upheld to explain the absence of perspective in the Arabic world, aniconism is the most common cultural history have ever delivered [START_REF] Grabar | The Formation of Islamic Art[END_REF][START_REF] Critchlow | Islamic Patterns: An Analytical and Cosmological Approach[END_REF][START_REF] Allen | Aniconism and Figural Representation in Islamic Art. Five Essays on Islamic Art[END_REF][START_REF] Callataÿ | Aniconisme et illustrations scientifiques en Islam[END_REF]. Strictly speaking, the word 'aniconism' refers to the prohibition of images making in Islamic tradition 11 . Nevertheless, this concept is unable to solve the problem we are faced with. This will be clear after some details will be given.

1. Arabic word for image (sûra) has multiple meanings. The verb sawwara means 'to shape, create' and subsequently 'to depict, represent'. And the question of aniconism in Islam is closely linked to the fact that the same word musawwir can refer to 'God' or to a 'painter' in line with religious or secular situations. Sûra always implies the idea that something has been composed or created (hudûth), but the word is equally multifarious. It can be rendered into 'shape, form, image, figure, silhouette, or face', according to the context. And that is why some texts are very confusing. The hadîth : "The house (al-bayt) in which there are images (al-tamâthil)… will be fled away by the Angels of Grace" (Bukhârî, Bad' al-Khalq, b. 17), is in fact compatible with both permissive and restrictive exegeses. The permissive interpretation understood al-bayt as 'temple' and al-timthal as 'carved image', while Ibn

Hanbal, in his restrictive view, read 'house' and 'image' respectively. Even if we consider that the ban was concerning mostly the depiction of animated being as Allâh, the Prophet, and all the creatures having a soul (rûh), the semantics remains a source of dispute. To begin with, when the Mekkans built anew the Ka'ba after its destruction by fire, they painted the pillars with images (suwar) of Prophets, Angels, and trees. Muhammad came then and ordered all the images to be erased, except those of the Virgin and Jesus (Wensinck, 1999: 927). Some We find here a first source of variation.

2. The prohibition of images is not based on Qur'ân directly, but on hadith literature.

The Qur'ân is in fact almost mute on the point to know whether the images are forbidden or not [START_REF] Besançon | L'Image interdite. Une histoire intellectuelle de l'iconoclasme[END_REF]. The absence of qur'ânic injunction joined to the split of Islam into various movements explains the deep impact of exegesis onto the point. Islam is not a single stream: it experienced areas, branches, legal schools, sects, and factions. The different movements applied different precepts concerning representation. Among the main branches of Islam, Shi'ites tolerates images much more than Sunnites and Kharijites ever did. Among Sunni legal schools, Hanafite madhhab has been much more receptive than Mâlikite, Shâfi'îte and Hanbalite. This is due to their respective position to personal reasoning (ra'y), personal effort (ijtihâd), and reasoning by analogy (qiyâs). Hanafite school admit both and, as it recognizes a form of self-determination, it is considered as the most liberal school. Mâlikite also admits the three criteria, but aggregates scholars' consensus (ijmâ'), a manner of curbing individualism, while admitting any innovation satisfying general interest. Shâfi'îte madhhab renounces to ra'y and ijtihâd, and it is thus responsible for the fossilization of ijtihâd devoid of its individual component; Hanbalite rite, opposed to Mu'tazilite rationalism, precludes both of three, it is thus the most rigorist school-the Wahhabite or Salafist movements proceed from.

3. The attitude regarding images varies according to the native cultures the Arabs conquered (the Roman, Byzantine, or Sassanid substrata, for instance). When the natives had a representational culture established for long, Islamic civilization took their features over.

And this is why Empire peripheries were often less reluctant to representation than the centre. Any look on the semantic ambiguities, theological divisions, and ethnical substratum thus shows attitudes covering the entire spectrum from iconophily to the most rigorist aniconism. Many artworks can illustrate the presence of iconic representations in Islam. 12 There is also a testimony in Ibn al-Haytham's Kitâb al-manâzir, Book II Chapter 3, and Book III Chapter 7, where he embarks on a detailed analysis of beauty (Sabra, 1989). Considering beauty (husn 13 ) among visual qualities (al-ma'ânî al-mubsara), he examines all the combinations of qualities that affect visual perception. His analysis is not restricted to the beauty of natural world. In several passages, he unambiguously speaks about the beauty of paintings and sculptures (nuqûsh wa tazâyîn).

Being iconic paintings favoured in some areas, and beauty conferred to them, aniconism cannot explain the distinctive features of Islamic art as a whole. We must here partake the view that "the categories that have traditionally been used to describe Islamic art such as 'aniconic' and 'nonrepresentational' are unhelpful for the problem at hand" (Gocer, 1999: 685).

But there is another argument to be pointed out. Had the images been forbidden, perspective pictures should have had all the chances to escape the interdict. Being a mode of depiction of carpentered worlds, as buildings and furniture, perspective is not directly concerned by the question of creatures. This point is for instance confirmed by the decoration of the Great Umayyad Mosque in Damascus (705-715). On the prayers hall façade, west portico, and treasure, several mosaics represent palaces and cities, decorated with branches, but without anybody inside. Rules of linear perspective could evidently apply to those representations, although they did not. Finally, the religious factor does not explain the iconic restriction in Islam, but only the non-homogenous distribution of representations in space and time.

The Axiological Factor: Overt Antirealism

The renewal of contemporary sociology greatly benefited from Boudon's contribution on the cognitive rationality and from the attention he paid to axiological motives, especially in the artistic domain (Boudon, 1995ab). He crossed the gap between conventionalism and Platonism, drawing his theory from 'axiological rationality' (Weber, 1956). This is an ideal framework for tackling the problem anew. What is indeed apparent in Islamic art, is the ambivalent attitude regarding hadith-and to follow or to get around a prescription is a sign of a value-based behaviour. To illustrate such ambivalence, we can evoke the opposition between the antirealistic features defining the major Islamic iconic artworks, and the competing theory of realistic illusionism developed in the optical literature.

As we said, in his Kitâb al-manâzir, Ibn al-Haytham embarks on the question of illusionism in painting (al-suwar al-musawarra). In this account, he appears to be a good witness of painters' ambiguous attitude about representation:

"Painters make their pictures and paintings look like the visible bodies to which they correspond, and by means of flat pictures they represent particular animals, individuals, plants, utensils or other solid objects, and their features. For this purpose they make skilful use of colours and drawings, paying particular attention to points of resemblance. For example, when they make pictures of hairy animals, fuzzy plants, rough-surfaced leaves or visibly coarse bodies, they make them look like the visible roughness of the surfaces of those animals or plants or inanimate bodies by means of drawings, outlines and different colours, though the pictures they make are flat and smooth or even polished. They also make pictures of individual people, imitating what is visible in their forms of the outlines of their faces and bodies, their hair, the pores and wrinkles in their skin… Painted pictures will be perceived to be like the forms they represent if those who made them were skilled in the art of painting." (Sabra, 1989, 1: 295, italics mine).

On the other hand, it is hard to find paintings conform to Ibn al-Haytham's statement.

Any survey of iconic paintings would show, by contrast, that realism is always hampered and knowingly corrupted. The most accurate view onto the point is that "the artist can get around the prohibition if he exhibits that he does not want to imitate the reality" (Papadopoulo, 1976: 55). Islamic iconic art is thus ascribed to a specific aesthetical value: the "principle of unlikelihood". In frontal projection as "Prince and Attendants", from a Dîvân by Jâmî (Plate 1A), and in oblique projection as "Ulugh Beg with Ladies of his Harem" from a Timurid Book (Plate 1C), there is a propensity to erase relief, shadows, spatial coordination, and point of view 14 -a skilful tactic to get around the harâm.

1. Shadow is only rarely cast on objects or individuals, except in some late Mughal miniatures of the XVI th century as the scene "The Loading of Gold" from the Book of Babur (New Delhi, National Museum, Visual Arts Gallery). This lack of shadowing could be a reference to Sahîh's commentary, where Nawawî generalizes the opinion of some ancient jurists, that "only reproductions that cast a shadow are forbidden" (Wensinck, 1999: 926). It is a manner to strengthen the difference between painting and sculpture, claiming that lawful plane pictures are not to be confused with illicit three-dimensional idols.

2. Relief is seldom treated in painting, except perhaps in some Ottoman miniatures as "The Nomad Camp" by Siyah Qalem (Istanbul, Topkapı Sarayı Müzesi). Insofar as relief, depth, drape, etc. imitate three-dimensionality (al-tajassum) of the depicted objects, they come under the previous case. The XI th century's Monophysite Yahyâ b. Jarîr writes: "As to prevent the faithful take them as idols, the images were painted flat and non corporal" (Papadopoulo, 1976: 58). Here again, painters' tactic could have been to enforce the distinction between volumetric and flat images, in order to escape the iconic proscription.

3. Point of view is always missing. Despite their names, frontal and oblique parallel projections differ much from isometric parallel projection. In isometric projection, the parallels result from the fact the point of view is situated at infinity. By contrast, there is no point of view in frontal and oblique projections, because they are pure mathematical conventions: it is not possible to see at the same time the front side without deformation, and the lines emerging perpendicularly to it. There is various possible explanations to the parallel projection choice, 15 but it is not dubious that it fits-better than any other system-with antirealistic values.

4. Spatial coordination of pictorial elements is usually lacking for, in the same scene, the painter could represent different objects according to various systems. Some rare exceptions happened in Central-Asian painting. We can cite the case of "Bahrâm Gûr in the Turquoise-Blue Pavilion", a scene from the Khamsa by Nizâmî (Plate 1B), and "The Feast of 'Îd" from a Dîvân by Hâfiz (Plate 1D). This feature can be understood as an implementation of 'atomism', taken either as a metaphysical outcome (Massignon, 1963, I: 12-13)

Conclusion

We have thus to solve the puzzle in a way that differs much from the specific solution foreseen by Weber (1922), that is nonetheless in keeping with the concept of axiological rationality. The idea that linear perspective arose only in the West due to the strength of an unusual process of rationalization is denied by the fact that IX th century Islamic scholars had yet a thorough knowledge of the optical and geometrical materials required in perspective. In addition, the process of rationalization was rarely so intense as in that time, because truth uniqueness and scientific communalism were core values of Medieval Islam. 17 The perspective puzzle is not a matter of less or more rationality, but a matter of axiological motives. First, Islamic disregard for perspective is the result of a value-set insofar painters were fearing the hadîths' prohibition, or wished to avoid pretentiousness of their name (musawwir). Second, Islamic art did not draw from Qur'ân and hadîth's precepts directly.

Third, painters implemented axiological motives, to put iconic practice in accordance with aniconic leanings. As a result, it is clear that such overt antirealism could be nothing but the main obstacle to the rise of linear perspective in Medieval Islam.

Finally, the question of Islamic disregard for practical perspective provides a shortcut towards the classic study of Renaissance perspective. I have argued elsewhere that Franciscan sensualism-realism-played a key role in the enhancement of perspective in trecento and quattrocento Italy (Raynaud, 1998: 291 sq). The multifaceted question of the rise of perspective cannot avoid an explanation by the aesthetic values: the call for realism makes clear the search for linear perspective in Late Medieval Italy, as overt antirealism elucidates the lackadaisical attitude of Islam regarding perspective. are defined by outline and contours, not by relief and volume… Line dominance meets with a linear, non spatial, organization of the composition" (Gonzalez in Beaugé and Clément, 1995: 72-73). 15 To begin with invariants related to each type of geometrical transformation. Eight basic invariants are known: distance (size, length), shape, area, angle size (perpendicularity), parallelism, collinearity (betweeness, straightness), ratio of division, cross-ratio (harmonic division). There are : 2 for projective transformation (crossratio, collinearity), 4 for affine projection (+ ratio of division, parallelism), 6 for similarity projection (+ angle size, shape), and 8 for metric projection (+ distance, area) (Hagen, 1989: 105). Suppose now-alongside Hagen's purposes-these properties actuate as a basis for the practical operations involved in the construction of each mode of depiction, thus projective transformation, i.e. linear perspective, suffers a severe disadvantage against other types. as the desert (or the sea), he must develop an individual knowledge of the stars in order to fix a precise azimuth; hence the supremacy of stars upon constellations, a dominance that ties up partly with the cognitive distinction between 'conceptual' cartographic representations, and 'practical' sequential representations. Such routines could have applied to space perception in general, considering the parts as more relevant than the whole. It is exactly the case of Islamic painting, that grants autonomy to objects within the pictorial composition. 17 "Islam provides a whole set of fundamental values. Among those values, one finds the uniqueness of truth, the lack of contradiction between revelation and reason, and thus between the two types of knowledge that they produce, the equality of human beings in jure before the truth and in the search for it, the pursuit of knowledge as a means to strengthen one's faith and as a form of prayer, the obligation to communicate knowledge and not to keep it to oneself, etc. These values, among others, have without the least doubt pushed forth research and have fostered creation of open scientific communities." (Rashed, 2003: 1).

Sources

  ; (2) Among other optical works, al-Kindî wrote a Book on the Rectification of the Errors and Difficulties dues to Euclid in his Book of Optics (Kitâb Abî Yûsuf Ya'qûb b. Ishâq al-Kindî ilâ ba'd ikwânihi fî taqwîm al-khata' wa al-mushkilât allatî li-Uqlîdis fi kitâbihi al-mawsûm bi-al-manâzir). Composed before 866, this treatise is important to apprehend the shift between al-Kindî's optics and the Euclidean tradition, to which he nevertheless pertains (Rashed, 1997); (3) Qustâ b. Lûqâ was a friend and a collaborator of al-Kindî. He also undertook studies in optics, and wrote a treatise entitled The Book on the Causes of the Difference of Perspectives that appear in the Mirrors (Kitâb fî 'ilal mâ ya'ridu fî al-marâyâ min ikhtilâf al-manâzir). This treatise is searching out in the direction of a geometrical-physiological synthesis. Being dedicated to al-Muwaffaq (229-78 H. / 843-91), it is posterior to al-Kindî's works, possibly ca. 870 (Rashed, 1997); (4) Finally, no study can ignore Ibn al-Haytham's Kitâb al-manâzir. In his masterpiece of optics, Ibn al-Haytham champions intromissionism and sensorial image (sûra) against Euclidean visual ray theory and Ptolemy's emissionism. Being included in the List III transmitted by Abî Usaybi'a in his

  Fig. 1 Fig. 2

  Commentaries. *,*** Although many scholars credit Ibn al-Haytham with the invention of the camera obscura (al-bayt al-muzlim), al-Kindî already experimented the rectilinear propagation of light with it. But, being equipped with a large aperture UZ, al-Kindî's camera obscura could only provide with fuzzy, not stigmatic, images onto the screen HT. Unlike his predecessor, Ibn al-Haytham stressed that the narrowing of the aperture reduce the divergence of rays(Sabra, 1989, 1: 17, Prop. I, 3 [14]). It is however unsure whether the instrument was rigouroulsy stigmatic. Camera obscura system is of a keen interest for perspective, because it paves the way to via instrumentalis, in which a picture can be drawn without geometrical reasoning. The apparatus by which Brunelleschi presumably depicted the first perspective view in Florence was of this type. Many other Renaissance scholars took up the apparatus on their own, asVinci (1970: 44-45), Cardano (1550, IV: 107),Barbaro (1568, V: 192),Danti (2003: 433), etc. ** Sabra is right when he writes: "there is no description in I.H.'s Optics of a picture obtained by means of a narrow opening… Kamâl al-Dîn, by contrast, has reported observations of reversed coloured images (suwar) of clouds and birds on an unlighted white surface placed behind a narrow opening Tanqîh, II: 399" (Sabra, 1989, 2: li). But this contrast gains not to be inflated. The image reversal is in fact contained in al-Kindî's and Ibn al-Haytham's records. In the first experiment, the rays issued from points B and G go to points K and L, and cross each other through the first screen. In the second experiment, when the star moves along BG [on the previous diagram], its image moves upside down along KL. There is again a striking continuity of aims and scope from Arabic optics to Renaissance perspective.

  So speaks Ibrâhîm b. Sinân in his Risâlat fî al-ma'ânî al-latî istikhrajahâ fî al-handasa wa al-nujûm (Epistle on the Examination of Difficult Calculus on Practical Geometry and Stars) 9 . And so speaks al-Bîrûnî in his Kitâb tastîh al-suwar wa-tabtîh al-kuwar (Book on Plane Projection of Constellations and Countries) 10 . Other crafts were in a very different situation. The mathematician Abû al-Wafâ' frequently complains about the lack of accuracy of the solutions provided by the artisans. Regarding their geometrical trial and error, he says:

  put the emphasis on the erased images, other on the remaining ones. The ambivalence repeats in Medieval doctrines, the Persian Abû 'Ali al-Fârisî (901-979), linked to Mu'tazilite rationalism, and the Andalusian al-Qurtubî (d. 1273) took position in favour of images, while Mujâhid Ibn Jabr (d. 722) and al-Ghazâlî (1058-1111) pronounced themselves against images.

  The combination of factors explains quite well the non-homogenous distribution of representations in space and time. The Umayyads, Nasrids, Fatimids, Ottomans, Ghaznavids, Saljûqids, Tîmûrids, Safawids, Qajars, and Mughals favoured depiction at a point defying what ever could imagine the Almohads, Almoravids, or Ayyubids. The aggregation of the previous factors brings to light a geographical contrast: Maghrib-apart al-'Andalûs-, East Africa, and Arabian peninsula mostly receptive to abstraction, in particular to the geometrical patterns used for mural tessellations and roof woodcarving vs. Near and Middle East, Turkey and Central Asia basically iconophilic.

  Alberti, L.B. (1992) De la peinture / De pictura (1435), ed. J.-L. Schefer, Paris: Macula. Amir-Moéz A.R. (1963) "A Paper of Omar Khayyam", Scripta Mathematica, 26: 323-337. Archimedes (1970) De la sphère et du cylindre, La mesure du cercle, Sur les conoïdes et les sphéroïdes, ed. Ch. Mugler, Paris: Les Belles Lettres.

  Here al-Kindî and Qustâ b. Lûqâ deal with (Euclidean) fundamental law of perspective, that an object becomes smaller and smaller as it recedes from the eye. * Al-Kindî proves the property geometrically. He establishes that equal lines GF, IH, LK, DC, cut out different angles of vision, according to the distance of the eye. Qustâ b. Lûqâ goes beyond Kindî because, if the angle tends to zero, the object reduces to a point, and remaining perpendicular to the axis of vision E, it will keep the axis. But a point belonging to the axis of vision is nothing but the 'vanishing point' (punto centrico, punto principale) in Renaissance perspective. Even if the concept does not appear in Arabic optics, everyone can verify the striking similarity that exists between Qustâ's reasoning and diagram, and those of Piero della

	Francesca (1984: 73-74, prop. XI), Leonardo da Vinci (1970: 35, 59-61), and Danti (2003:
	194, th. XXVII).

this basic result. ** He considers that a single object, AB, moved away from the eye, takes the successive positions GF, IH, LK, DC. Thinking a continuum, Qustâ had no reason to stop at DC. *** He thus considers AB is moving away to infinity. This makes a notable difference with al-

  , and although geometrical optics was cultivated at the highest level-including the very sources of what will become the Renaissance standard-, linear perspective did not arise in Medieval Islam. Other types of representation were preferred as parallel frontal and oblique projections. Frontal projection underlies for instance scenes as "Prince and Attendants", from a Dîvân by Jâmî illustrated under the Safawids (Plate 1A), or "Bahrâm Gûr in the Turquoise-Blue Pavilion" from a Khamsa by Nizâmî, under the Safawids (Plate 1B). Oblique projection underlies scenes as "Ulugh Beg with Ladies of his Harem" from the Timurid Samarqand (Plate 1C), or "Feast of 'Îd" from a Dîvân by Hâfiz, under the Uzbek dynasty (Plate 1D). Frontal and oblique projections were also used in scientific and technical literature, for example al-Jazarî XII th

century compendium Kitâb al-jâmi' bayna al-'ilm wa al-'amal al-nâfi' fi sinâ'at al-hiyal (Book of Theory and Useful Practice in the Mechanical Devices) (New York, Metropolitan

  Plate 1.-1A "Prince and Attendants", Dîvân by Jâmî (Safawids) S 1986.49.-1B "Bahrâm Gûr in the Turquoise-Blue Pavilion", Khamsa by Nizâmî (Safawids) F 1908.275a.-1C "Ulugh Beg with Ladies of his Harem", in a book from Samarqand (Timurids) F 1946.26.-1D "The Feast of 'Îd" from a Dîvân by Hâfiz (Uzbeks) F 1932.51. Courtesy of Freer and Sackler Galleries, © Smithsonian Institution, Washington D.C.

	, or as a
	practical leaning that influenced Islamic civilization-the Bedouin nomadic form of life
	facilitating a sequential way of thinking.

16 1A. Uncoordinated frontal projection 1B. Coordinated frontal projection 1C. Uncoordinated oblique projection 1D. Coordinated oblique projection

  16 IbnKhaldûn (2002) pioneered the study of opposite forms of life of Bedouin civilization (al-'umrân albadawî) and sedentary civilization (al-'umrân al-hadharî), but deals with painting only rarely. The Bedouins influenced notably scientific practical procedures. For instance, while Greek astronomers divided the sky into constellations, Arabic astronomers individuated the stars by a proper name (al-dabarân 'the Follower [of the Pleiades]' for α Tauri; al-khibâ' 'the Tent' for α Corvi; al-mufrid 'the Solitary One' for η Bootis, etc.) This is logical according to Bedouins (or Sailors) fundamental leanings. If one has to orientate himself in an open space

 Saliba G. (1999)"Artisans and Mathematicians in Medieval Islam", Journal of the American Oriental Society, 119 (4): 637-645. Soucek P.P. (1999) "Taswîr", Encyclopédie de l 'Islam, vol. 10, Leiden: E.J. Brill, 387-390. Weber M. (1922) Gesammelte Aufsätze zur Religionssoziologie, Band I, Tübingen: Mohr. Weber M. ( 1956) Wirtschaft und Gesellschaft, Tübingen: Mohr. Wensinck J. (1999) "Sûra", Encyclopédie de l'Islam, vol. 9, Leiden: E.J. Brill, 925-928.

* Université Pierre-Mendès-France (Grenoble), GEMAS (Paris). 1 Kitâb Uqlîdis fî ikhtilâf al-manâzir, Prop. 42 (Kheirandish 1999: 146-147).

2 Kitâb fî taqwîm al-kha ta' wa al-mushkilât allatî li-Uqlîdis fî kitâbihi al-mawsûm bi-al-manâ zir, Prop. 38 (Rashed, 1997: 272-275).

3 Kitâb Uqlîdis fî ikhtilâf al-manâzir , Prop. 11 (Kheirandish, 1999: 34-37). I have changed the lettering according to al-Kindî commentary, for the sake of comparison. 4 Kitâb fî taqwîm al-khata ', Prop. 11 (Rashed, 1997: 190-195). 5 Kitâb fî taqwîm al-kha ta ', Prop. 6 (Rashed, 1997: 178-181). I have introduced little changes in the lettering in order to compare this proposition to the next one, by Qustâ b. Lûqâ. 6 Kitâb fî 'ilal mâ ya'ridu fî al-marâyâ min ikhtilâf al-manâzir, Prop. 4-5 (Rashed, 1997: 582-585). 7 Liber Jacob Alkindi de causis diversitatum aspectus et dandis demonstrationibus geometricis super eas , Prop. <6> (Rashed, 1997: 448-450). The last sentence comes in fact in the intercourse of the passage. 8 Kitâb al-manâzir, I, 3 [2, 4] (Sabra, 1989, 1: 13, 14). 9 "I liked his craftsmanship (sun'atahu) and found him to be technically clever (latîf al-hîla) in his work. So I dictated to him the description of the instrument, demonstrating how to set up on a spherical surface a gnomon whose shadow falls on the spherical surface during the whole day for all days of the years. I made my description for him of the type that befits those who work with their hands (al-sunnâ' al-ladhîna ya'malûn bî alyad)" (Saliba, 1999: 641-642). 10 "Some of the artisans favour the use of arithmetic and prefer it over craft methods (yu'thiruhu 'alâ al-turuq alsinâ'iya), as we have found with all the makers of astrolabes and instruments" (Saliba, 1999: 642). 11 The thesis of a Jewish origin of this aversion is unclear. On the one hand, the aniconism expressed in the second commandment, Exod 20:4-6, as in Exod 32:1-4, Lev 26:1, Deut 4:15-18, Deut 5:8-9, Deut 27:15, and many Talmudic commentaries, is counterbalanced by Exod 25:18-20, Num 21:8-9, Ezech 1:26-28, II Chron 3:10-13, II Chron 4:3-4. On the other hand, Jewish art has always been submitted to iconic-aniconic similar tensions. See for instance the iconic paintings at the synagogue of Doura Europos (244-245) vs. the aniconic decoration of Samuel Halevi Abulafia's synagogue in Toledo (1357); the iconic miniatures by Isaac Sofer b. Obadia for Jacob b. Asher's Arba a Turim (1435) vs. the aniconic carpet-page micrographies of the Bible of Burgos by Menahem b. Abraham b. Malik (1260), etc. [START_REF] Gutmann | The Second Commandment and the Image in Judaism[END_REF][START_REF] Klagsbald | A l'ombre de Dieu[END_REF]. 12 We can cite: the fountains of Madînat al-Zahrâ's palace near Córdoba (Spanish Umayyads); the fountain of Lions of the Alhambra in Granada (Nasrids); coins with the head of caliph 'Abd al-Malîk (685-705) (Umayyads); a lion hunting gazelles under a pomegranate tree, Khirbat al-Mafjar winter palace, Jericho valley, 724-743 (Umayyads); scenes of building, fishing, bathing, music playing and dancing, including naked girls, on the walls of Qsar 'Amra palace, Jordan, VIII th century (Umayyads); dancers on mural paintings in Dâr al-Khalîfa at Sâmarrâ', Iraq, IX th century ('Abbâssids); a 'ud player on a carved panel of ivory, Egypt, XI-XII th century (Fatimids); fortresses and branches on the walls of the mausoleum of Baibars the I st , Damas, 1277 (Mamluks); a rabbit on a dish from Tell Minis, Syria, XII th century (Saljûqids); countless lion incense-burners, Khorasan, XIIth century (Ghaznavids); a princely assembly painted on the walls of Lashkar-i Bâzâr, Afghanistan, IX th century (Ghaznavids); the many miniatures of Shâh nâma by Fîrdawsî (Ghaznavids), Kalîla wa Dimna by Ibn al-Muqaffa' (Tîmûrids), Zâfar nâma by Sharaf al-Dîn Yazdî (Tîmûrids), and especially the copy of Nizâmî's Khamsa for Amîr 'Alî Farsî Barlas, that even depicts the face of the Prophet in his mi 'râj, prob. Herat, 1494-95 (Tîmurîds); two fighters in a page of Sâ'di's Gulistân, Bukhara, XVI th (Shaybanids); the animals that decorate the façade of Shîr Dâr madrasa in Samarqand, 1619-1635 (Janids); the widespread practice of muraqqa', i.e. bound collections of calligraphies and paintings, under various dynasties (Safawids, Ottomans, Mughals)… [START_REF] Ettinghausen | The Art and Architecture of Islam[END_REF][START_REF] Blair | The Art and Architecture of Islam, 1250-1800[END_REF][START_REF] Barrucand | Les fonctions de l'image dans la société islamique du Moyen-Âge[END_REF][START_REF] Gonzalez | Réflexions esthétiques sur l'approche de l'image dans l'art islamique[END_REF]. 13 The root hasana includes moral connotations that jamala has not. 14 "From the aesthetic point of view, the plastic processes that provide those images tend to dematerialize them, and give them an ornamental nature rather than a descriptive one… Figures and animals come into existence and