M Gouiffès

) C Collewet

) C Fernandez-Maloigne

) A Trémeau

A PHOTOMETRIC MODEL FOR SPECULAR HIGHLIGHTS AND LIGHTING CHANGES. APPLICATION TO FEATURE POINTS TRACKING

This article proposes a local photometric model that compensates for specular highlights and lighting variations due to position and intensity changes. We define clearly on which assumptions it is based, according to widely used reflection models. Moreover, its theoritical validity is studied according to few configurations of the scene geometry (lighting, camera and object relative locations). Next, this model is used to improve the robustness of points tracking in luminance images with respect to specular highlights and lighting changes.

INTRODUCTION

Most computer vision applications based on the computation of correspondences between images are sensitive to illumination changes. Indeed, the robustness of image processing with regard to such a phenomenon remains a crucial problem. Moreover, in luminance images, no global compensation of illumination changes has been defined as far now, contrary to color images [START_REF] Gevers | Object recognition based on photometric color invariants[END_REF]. In general, luminance is supposed to be constant in a small window of interest W during an image sequence [START_REF] Horn | Determinig optical flow[END_REF], but that is correct only for lambertian objects viewed under perfectly constant lighting. Photometric normalization, as in [START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF] for example, or local photometric models answer partially this issue. The affine photometric model has been widely used, for example in optical flow [START_REF] Negahdaripour | Revised definition of optical flow:integration of radiometric and geometric cues for dynamic scene analysis[END_REF] or tracking [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] applications, but the parameters of this model are supposed to be spatially constant in each point of W. Moreover, it has been shown in [START_REF] Gouiffès | Feature points tracking : robustness to specular highlights and lighting changes[END_REF] that this model is based on the same assumptions as the photometric normalization. In [START_REF] Black | Robustly estimating changes in image appearance[END_REF] however, the spatial variations of illumination are taken into account. Nevertheless, the assumptions on which these local models are based (according to the physical properties of materials and to the scene geometry) have not been clearly specified, and the definition of each of their parameters has not been given. In this article, we propose a local photometric model adapted to specular highlights and lighting changes. We analyze clearly the assumptions on which it is based, according to the reflection models that are widely used in computer vision and computer graphics. Our second contribution consists in studying theoritically the validity of the proposed model according to some borderline geometry conditions, such as the position of the camera and the light source. To finish, we focus on feature points tracking. The proposed photometric model is used to improve robustness of points tracking with respect to specular highlights and lighting changes, the parameters of the photometric model are computed simultaneously with the parameters of the motion model.

This article is structured as follows. Section 2 focuses on the modeling of luminance changes. Then, the model of photometric changes and its local approximation are proposed in section 3. The section 4 is dedicated to the study on the model validity. To finish, section 5 exposes the tracking method and shows some results obtained on real image sequences.

LUMINANCE MODEL

Let us suppose that f and g are respectively the images of an object acquired at the two different times k and k ′ . A point P of this object projects into image f in p of coordinates (x p , y p ) and in p ′ of coordinates (x ′ p , y ′ p ) into image g, after a relative motion between the camera and the scene. The luminance at p depends on the scene geometry. Fig. 1 describes the vectors and the angles used in this article. V and L are respectively the viewing and the lighting vectors, which form the angles θ r and θ i with the normal n in P . B is the bisecting line between V and L, it forms an angle ρ with the normal n. According to the most widely used reflection models, such as the Torrance-Sparrow [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF] and the Phong [START_REF] Phong | Illumination for computer generated images[END_REF] ones, the luminance at p can be described as follows

f (p) = K d (p)a(p) cos θ i (P ) + h f (p) + K a (1)
where K a is the intensity of ambient lighting and K d a diffuse coefficient corresponding to the direct lighting intensity. These values depend also on the gain of the camera. The term a(p) is related to the albedo1 in P . Function h f expresses the contribution of the specular reflection, which vanishes in case of a pure diffuse reflection, that is for Lambertian surfaces.

Consequently, with such objects and for a given lighting direction L, the luminance at p is the same whatever the viewing direction V is. For non-lambertian objects, h f reaches its maximum value when ρ(P ) = 0, that is when B coincides with n (see for example [START_REF] Phong | Illumination for computer generated images[END_REF][START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF]).

LOCAL PHOTOMETRIC MODEL

Now let us consider that some illumination changes expressed as functions t a , t d and t i are respectively provoked on K a , K d and θ i . Therefore, after a lighting change (intensity or location), the luminance g at time k ′ becomes

g(p ′ ) = K ′ d a(p) cos θ ′ i (P ) + K ′ a + h g (p ′ ) (2)
with 1) and ( 2), the photometric changes become

K ′ d = K d + t d , θ ′ i (P ) = θ i (P ) + t i (P ) and K ′ a = K a + t a . According to (
g(p ′ ) = λ(p)f (p) + η(p) (3)
where λ(p) and η(p) are given by

λ(p) = K ′ d (p) cos(θ i ′ (P ))/K d (p) cos(θ i (P )) (4) η(p) = -(h f (p) + Ka)λ(p) + hg(δ(p, A)) + Ka + ta (5)
with θ r and 3) has also been obtained in [START_REF] Black | Robustly estimating changes in image appearance[END_REF]. However, to our knowledge, no formal definition of λ and η had been established.

θ i ∈]-π 2 π 2 [. Equation (
Let us assume that the surface around P is described by a C 1 function. Consequently, when the light source is sufficiently far from the surface then the angle θ i and the function t i vary in W in a continuous way. The direct light intensity K d is also supposed to vary smoothly in W. In those conditions, λ is a C 1 function in W.

Now, let us assume that the surface projected on W has nearly the same roughness parameter in each point. For example, that is correct when each point of W is located on the same type of material. Then, by assuming a continuous surface projected on W, the functions h f and h g are continuous since they depend on the normal in each point of W.

Consequently, λ(m) and η(m) are continuous in each point of W and therefore can be expanded in Taylor series at first order around the center of W, p. Therefore (3) becomes

g(m ′ ) = U T λf (m) -g(δ(m, A)) -U T η (6) with U = (1, x -x p , y -y p ) T , λ = (λ 1 , λ 2 , λ 3 ) T and η = (η 1 , η 2 , η 3 ) T .

VALIDITY OF THE MODEL

This section will focus on the study of the validity of [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF]. In order to simplify this problem, we consider a moving light source according to a motionless scene. This configuration enables us to study the lighting changes and the specular highlights occurrence. In this purpose we consider the geometry depicted by Fig. 2. We call P a physical point of coordinates (X p , Y p , Z p ) T expressed in a frame R c related to the camera. P projects in p, at the center of W. Point M , of coordinates (X, Y, Z) T , projects in m in the neighborhood of p in W. In addition, Π is the tangent plane of the surface in P .

When the surface around P is described locally by a C 2 function, it can be expanded in a Taylor series around P . Then the coordinates of M can be approximated by

Z = Zp + D X (X -X P ) + D Y (Y -Y P ) + D XX (X -X P ) 2 + D Y Y (Y -Y P ) 2 + D XY (X -X P )(Y -Y P ) (7)
where D X , D Y , D XX , D XY and D Y Y are the first and second derivatives of the surface in P with respect to X and Y . The normal vector in M is expressed by n = (D X , D Y , -1) T . We call S = (S x , S y , S z ) T the position of the direct light source and therefore we have L = (S x -X, S y -Y, S z -Z) T and V = (-X, -Y, -Z) T . B is the bisecting line between V and L and therefore it can be expressed with respect to (X, Y, Z). In the same way, the angles θ i , θ r , ρ, which depend on n, V , L and B, can also be expressed with respect to X, Y, Z. Consequently, λ(m) and η(m) can be expressed with respect to X, Y, Z. Then, we assume that both the camera and the scene are motionless between k and k ′ , and that the direct light source undergoes a small motion dS = (dS X , dS Y , dS Z ) T from its initial location S. After a perspective projection of M to point m of coordinates (x, y), λ(m) and η(m) are expanded in Taylor series around p (center of W). At second order, we obtain

λ(m) = λ 1 + λ 2 x + λ 3 y + λ 4 x 2 + λ 5 y 2 + λ 6 xy (8) η(m) = η 1 + η 2 x + η 3 y + η 4 x 2 + η 5 y 2 + η 6 xy (9)
The model [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] assumes that λ 4 , λ 5 , λ 6 , η 4 , η 5 , η 6 are null in [START_REF] Phong | Illumination for computer generated images[END_REF] and [START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF], but what does it mean in a geometrical and physical point of view? To answer this question, we shall determine the configurations of acquisition for which the model is the most adapted. Therefore, λ i and η i for i = 4..6 are expanded in Taylor series at first order around dS ≃ 0. We call λ i and η i these approximations. We only analyze the expansion of λ 4 and η 4 , since almost the same conclusions arise from λ 5 , λ 6 and η 5 , η 6 .

• Validity of λ. Let us consider three interesting configurations.

1) When the lighting and the normal vectors are convergent L = αn it yields to λ 4 = 0. Thus, a small motion of the light source does not infer on the coefficients of second order of [START_REF] Phong | Illumination for computer generated images[END_REF].

2) When the light source is close to the camera S = O, we obtain b

λ 4 = 2(D Y D XX + D XY D X )dS Y -(4D XX + 2 Zp )dS Z + " 6D X D XX -2D X Zp " dS X .
We can see from this relationship that λ 4 ≃ 0 when the surface is nearly planar (D XX ≃ 0, D XY ≃ 0) and the camera is far from the surface (high Z P ).

3) When the light source S is located at a very small distance ǫ with respect to P , the coefficients of λ 4 can be expanded in Taylor series around ǫ, and λ 4 becomes

b λ 4 = 2Zp(D X (ǫ-1)+D XX (2ǫ 2 +Zpǫ)) ǫ 3 dS X + 2Zp (D XX Zp D Y +D XY D X ǫ) ǫ 2 dS Y - 2Zp (Zp +D 2 X ǫ-Zp D 2 X +2Zp D XX ǫ) ǫ 3 dS Z
Since the small distance ǫ appears at the denominator, this is an unfavourable configuration to use the photometric model.

• Validity of η. In order to simplify the analysis, we consider λ(m) = 1 in (5), such that only specular highlights are supposed to be caused. Because h f and h g reach their maximum value when ρ = 0, we study the variations of η around this specific configuration, by using the Phong model. Therefore, the initial location of the light source S is chosen so as B coincide with n (ρ = 0). We note d the distance between P and S. Two configurations are studied:

1) First, when L = n = V , the tangent plane Π is parallel to the sensor plane (D X and D Y = 0) and θ i = 0. Since the relations obtained in that case are too complicated to deduce any conclusion, we consider two further constraints. Firstly, if the light source is initially close to the surface, then d is small and η 4 can be expanded in Taylor series at first order around d = 0 so that b

η 4 = -n " 2D XX + 1 Zp " dSZ.
According to this relationship, the modeling error is low when the surfaces are rough (n low) and almost planar (low D XX ).

Secondly, if the camera is near the surface (low Z P ) then η 4 ≃ 0. That is a favourable condition to use [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF].

2) Now, let us consider that the orientation of the tangent plane Π with respect to the sensor plane is low (low D X and D Y ). The higher this orientation is, the larger the angle θ i is. In that case, we expand each term of η 4 in Taylor series around D X and D Y . Because of the complexity of the relations obtained, the surface is assumed to be planar (D XX = 0 and D XY = 0) and two configurations are considered. Firstly, the camera is supposed to be located near the surface (d small), then

b η 4 = -nD X 4Zp (3n + 7) dS X -nD X 4Zp (n + 1) dsY -n Zp dS Z
According to this expression, the lower the orientation of Π is (low D X and D Y ), the more ( 6) is valid. Secondly, when the camera is close to the surface (Z p small), we obtain η 4 ≃ 0.

• Conclusion. Consequently, the approximations of λ and η by a Taylor series at first order are quite realistic (the terms of second order of λ and η are almost null) when:

the orientation of the tangent plane Π of the surface in P is low with respect to the sensor plane; the second derivatives D XX , D Y Y and D XY of the surface are low; -L coincide with n ; the camera is close to the surface (low Z P ) and the lighting source is far from the surface; the surface is rough (low n). Let us notice that the affine photometric model [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF][START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF] is far more restrictive than (6) because λ 2 , λ 3 , η 2 and η 3 are supposed to be null simultaneously in equations ( 8) and (9).

APPLICATION TO FEATURE POINTS TRACKING

We call δ the motion model of a small window of interest W centered on p. We assume that this motion is parametrized by a vector A so that p ′ = δ(p, A). The point tracking method consists in computing A and the photometric parameters λ and η (see equation ( 6)) by minimizing the following criterion

ǫ(A, λ, η) = X m∈W " U T λf (m) -g(δ(m, A)) -U T η " 2 (10)
In the following experiments, we choose an affine motion model, which is computed between the first frame and the current one. This technique, that we call M1, is compared on two image sequences (see Fig. 3a and3b) with the method proposed by Jin and Soatto, based on the computation of an affine photometric model [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF]. We call M2 this latter method.

In each case, the objects are motionless and the camera moves. The first sequence shows a book, the cover of which is made of glossy paper. Consequently, the motion of the camera induces some specular highlights variations. No lighting change is caused. This sequence is played from the first frame to the final one and then from the last one to the first one. The second sequence shows a specular object covered by a glossy paper. It is lighted by the daylight and a direct lighting, the intensity of which varies periodically from a low value to a high one each 20 iterations approximately. The points are selected by the Harris detector [START_REF] Harris | A combined corner and edge detector[END_REF] (52 points in the first sequence and 25 in the second one).

The tracking algorithm integrates an outlier rejection step, based on the analysis of the convergence of residues ǫ: a point is rejected as soon as its residues become greater than a threshold (here, a mean luminance variation of 15 is tolerated for each pixel of W). The size of the window is 15 × 15. The use of smaller windows can lead to a poor accuracy of the photometric and geometric parameters involved in M1. For small windows, it is more suitable to use the method proposed in [START_REF] Gouiffès | Feature points tracking : robustness to specular highlights and lighting changes[END_REF] which uses only 3 parameters. This latter approach is not used in this article since it is less efficient for large windows. In the first sequence, 18 points are lost by M1 and 34 by M2 (almost twice more points) because of the specular highlights variations. In the second one, only 1 point is lost by M1 and 4 by M2, because of the severe lighting changes. In order to compare the accuracy of the geometric and photometric modeling, Fig. 3c and3d show the evolution of the residuals (the average of the residuals computed on the points that are correctly tracked by M1 and M2) obtained by each technique with respect to the iterations, respectively for the first and the second image sequences. They are in adequacy with the geometric and lighting changes. Indeed, in Fig. 3c the residues are symetric. In Fig. 3d, the residues vary from a low value to a high one each 20 iterations. In each case, the residuals obtained by M1 are lower than those obtained by M2, which proves that the photometric model is always more adapted to the real illumination changes. In particular, M1 compensates better for specular highlights, as it is shown by the number of points correctly tracked in the first sequence and the residuals obtained. When convergence residuals are low, it is very probable that the accuracy of the points location is good. In addition, the computation times are not significantly increased (5.2 ms for method M1 and 4ms for method M2, on the first sequence).

CONCLUSION

This article provided a theoritical explanation of luminance changes and proposed a local photometric model, based on the study on widely used reflection models. We analyzed the validity of this model by considering a locally continuous surface viewed under some borderline configurations of the scene geometry. This model can be used in various computer vision applications based on the computation of correspondences between frames. In this article, it is introduced in a points tracking technique. Compensating the illumination changes during the image sequence improves the robustness of the tracking process when specular highlights and lighting changes occur. Indeed, a larger number of points is correctly tracked and the residuals of the method are low. However it could be interesting to study the accuracy of the points location during the tracking process. That will be part of our future works.

Fig. 1 .

 1 Fig. 1. Vectors and angles involved in the reflection description.

Fig. 2 .

 2 Fig. 2. Model of the scene geometry.

Fig. 3 .

 3 Fig. 3. (a) and (b): two frames of the image sequences. (c) and (d): Comparison of the residuals obtained with M1 and M2: on the first sequence (c) and the second one (d).

The albedo is the ratio of the amount of light reflected by a small surface in P to the amount of incident light. It only depends on the material and its texture.


Anonymi Cantabrigiensis commentarium in Aristotelis sophisticos elenchos, S. Ebbesen (ed.), Copenhagen, Det Kongelige Danske Videnskabernes Selskab, 2019. PROLEGOMENA BACKGROUND. In hindsight, it is not surprising that the exegesis of Aristotle's Sophistici elenchi developed into one of the most substantial corpus of the Latin commentary tradition. To make a long story short, in its customary capacity as the art of arts and the science of sciences, mediaeval logic was primarily concerned with discerning the true from the false in arguments as they occur in natural, ordinary speech as opposed to the more formalised parlance later logicians will resort to. It makes perfect sense then that mediaeval logicians paid special attention to everything liable to disrupt sound reasoning thus preventing us from speaking the truth. Indeed, they were second to none and better than most at exposing and elucidating arguments' flaws and shortcomings. After all, as John Buridan -faithful to a long and illustrious tradition -aptly put it, « rooting out errors » is logic's first order of business 1 . As early as the 1140s, Aristotle's Sophistici elenchi provided the most fertile ground for such keen interest in fallacies ; which, in turn, explains etc.

RELEVANCE. This much is uncontroversial or, at any rate, can withstand any amount of scrutiny we care to throw at it. Courtesy -first and foremost -of Sten Ebbesen, whose long-standing interest in mediaeval writings on bad arguments has turned the Byzantine and Latin aftermath of Aristotle's Sophistici elenchi into wellcharted territory -by all standards. As it happened, Anonymus Cantabrigiensis has played no small part in shaping this picture. As a matter of fact, time and again over the last forty years or so, quotes and insights from the anonymous work have kept showing up in Sten Ebbesen's editions and studies : since he first discovered the commentary in the late 1970s and brought it to the general attention, Sten Ebbesen has routinely drawn on the Anonymus as an early witness of the circulation of Aristotelian logical works and related texts, as a convenient illustration of major trends and distinctive features of the Latin literature on fallacies, as well as a sensible interpreter in his own right 2 . OPUS READERSHIP. Not only Anonymus Cantabrigiensis has been on Sten Ebbesen's radar from the beginning, but the commentary itself has circulated freely amongst his pupils and colleagues as early as August 2009 -it being, in all likelihood, the main if not the only reason why it took him about ten years to see it through the press 3 . Good things come to those who wait and all, the final result -a complete edition of the extant text (p. 53-376), preceded by a lengthy introduction (p. 9-50) and followed by an index locorum (p. 377-380) and an index verborum (p. 381-407) -is everything one can expect from a veteran editor of logical texts and a fine connoisseur of the commentary tradition.

INTRODUCTION. Sten Ebbesen's edition features a substantial introduction, which tackles both doctrinal and philological issues, in that order. A thorough survey of the early Latin commentaries on the Sophistici elenchi is conducted first. The Anonymus Cantabrigiensis' place within this tradition is discussed next. New internal and old anecdotal evidence is carefully weighed and, on account of the former being fully available for the first time, Sten Ebbesen's conclusions are quite different from those he previously reached, most notably in the pioneer essay on the origins of British logic, namely Ebbesen 1985. Anonymus Cantabrigiensis is still depicted as a conservative, old-school logician, but he is no longer hailed as a late twelfth-century Englishman having 1 . « <Logica> habet enim unam partem sophisticam quae est exstirpativa falsarum rationum [it belongs to logic to eradicate errors] » (Iohannis Buridani summulae logicales, Prooemium, 7.11-12) or so we read at the very beginning of Buridan's handbook of logic, where the elimination of false arguments provides the main ground for the commander metaphor John Buridan was fond of.

2 . A perfunctory background check will have to suffice for our present purposes. In addition to contributing a number of fragments to Sten Ebbesen's "Alexander" collection (Ebbesen 1981a, III p. 145, 149, 194, 244, 246, 259 with the additional item in Ebbesen 1990, p. 115), Anonymus Cantabrigiensis figures prominently in his reconstruction of the late XIIth and early XIIIth century Latin reception of Aristotle's Prior and, most notably, Posterior Analytics (cf. respectively Ebbesen 2010, p. 99 andEbbesen 2015, p. 17). Besides exemplifying both borrowings from traditional logical doctrine (e.g. the distinction between a syllogism's matter and its form, as recalled in Ebbesen 1981b, p. 6) and Latin innovations (like the « cause of appearance » vs the « cause of deficiency » device applied to the analysis of arguments, as expounded in Ebbesen 1987, p. 116), Anonymus Cantabrigiensis' views are studied for themselves in connection with topics as diverse as context-sensitive arguments [START_REF] Ebbesen | Context-Sensitive Argumentation. Dirty Tricks in the Sophistical Refutations and a Perceptive Medieval Interpretation of the Text[END_REF], ill-formed sentences (Ebbesen 1981c, p. 95-96) and issues with Aristotle's typology of disputations [START_REF] Ebbesen | Demonstrative Disputation -A contradictio in adiecto? Medieval and Recent Commentators on Aristotle's Sophistical Refutations, Chapter 2[END_REF].

3 . It is only fair to mention as a matter of record that Anonymus Cantabrigiensis' is neither the first nor the only commentary Sten Ebbesen has virtually edited and generously made available to people in or with links to the Copenhagen School of Medieval Philosophy [START_REF] Ebbesen | Doing Philosophy the Sophismatic way. The Copenhagen School, with Notes on the Dutch School[END_REF]. A provisional list of these in-all-but-name editions should include at least five more items in [START_REF] Ebbesen | Medieval Latin Glosses and Commentaries on Aristotelian Logical Texts of the Twelfth and Thirteenth Centuries. Appendix : A list of Latin Commentaries on the Sophistici elenchi and Treatises De fallaciis, c. 1125-1300[END_REF] taught for a while somewhere between Paris and Rouen before shipping back to England with his precious library. In fact, rather than one of the forefathers of the British logical tradition -as initially suggested by Sten Ebbesen himself showcasing the OXYNAT hypothesis (where, as one may recall, « "OX" stands for Oxford, "NAT" for native, "Y" for y ») -what we are looking at now is a learned Parisian master well past his novice days who has been there and done that long enough to bring into the next century a sound knowledge of the old Paris sects and a repeated involvement in lecturing on Aristotle's work on fallacies. Nostalgic readers might wonder whether the ancestral hero Sten Ebbesen had conjured back in the day deserved to fall into oblivion without further adieu ; be it as it may, no one will fault the portrait which has replaced it for lacking in depth and accuracy, for Sten Ebbesen paints it down to the last detail through a comprehensive assessment of Anonymus Cantabrigiensis' familiarity with the Latin translations of the Aristotelian corpus and his late ancient an byzantine commentators (a puzzling echo of Zeno's paradoxes against motion, as discussed in Physics VI 2 and 9, adds a little mystery to an otherwise conventional, albeit extensive, acquaintance with the usual sources), as well as a meticulous comparison between Anonymus Cantabrigiensis' views, on one hand, and the positions he was cognizant with from the time he was himself a young student on (as far as scholarly allegiances go, a residual affiliation to the « nominal » school is Sten Ebbesen's educated guess), on the other hand.

TEXT. The commentary survives in one codex only -Cambridge, St John's College D.12 (C) -which Sten Ebbesen describes from all sides and angles (C's history, fabric and layout, contents, etc.) relying on his own expert observation and the advice of fellow palaeographers Anne Grondeux, Dominique Poirel and Rodney Thomson. Insofar as emendation is the only way out of trouble when dealing with the oddities and flaws of a text which has been handed down in a single manuscript, Sten Ebbesen singles out confusing features of C's handwriting, orthographic and morphological habits of C's scribe, his proclivity to misread, omit and slip in any way, nature and extent of the textual corruptions such misgivings lead to as well as the soundness of the many and varied corrections they brought about on the fly or as an afterthought. And this is where Sten Ebbesen's consummate craftsmanship as a specialist of Aristotelian commentaries and logical texts stands out. As a case in point, one only has to refer to the way he has dealt with one the most common and yet trickiest features of the family of texts the anonymous commentary belongs to, namely the fact that C's text is extensively abbreviated -many abbreviations being, needless to say, virtually undistinguishable and, for the same reason, open to more than one interpretation : « Anyone not familiar with the abbreviation system of the time may find many of my emendations implausibly farfetched, but I have, in fact, refrained from emending if I could find no paleographically plausible path from my assumed original text to the one actually found in C » (p. 41).

As Sten Ebbesen's maxim makes it plain, neither ingenuity nor expedience -let alone whim and imagination -should guide editors caught between the conflicting imperatives of either following a single manuscript's readings to a fault or changing the facts of the text to fit the best sense one can squeeze out of it. Restoration (through the painstaking process of tracing back an error to its most probable cause) should be the editor's guiding star and the « paleographically plausible path » out of the text's conundrums what keeps him honest at every turn. Sten Ebbesen being Sten Ebbesen, the question whether he has followed his own advice is a rhetorical one, as demonstrated by a hundred footnotes where the evidence is presented and occasionally spelled out for the edification of laymen and specialists alike4 . AUCTOR A GLIMPSE INTO ANONYMUS C MIND : WEIRD QUESTIONS AND ASTUTE ANSWERS. As every teacher will tell you, there are no dumb questions, only dumb answers. While being no exception, « How come that Aristotle quoted Vergil's and Horace's verses as examples of his fallacy of accent ? » has nonetheless an odd ring to it … Moreover, the issue is not likely to shed much light on anything relevant or important either by itself or by proxy (except maybe for the identification of Boethius as the culprit) : after all, it is not so much the translations that advertised themselves as something else or looked mighty suspicious that got Latin commentators into any trouble worth mentioning. Still, some of them took Aristotle's baffling knowledge of Latin poets seriously enough to challenge the authenticity of the work that confronted them with the fun fact in the first place. In the words of the early Parisian glossae, which De Rijk 1962, I, p. 83 convincingly dated around the mid-twelfth century :

[T01] Anonymi glosae in Aristotelis sophisticos elenchos, I, 326.1-4 : « notandum est quod quidam ob hoc dicunt quod Aristotelem non fecisse Elenchos, quia non exempla graecorum, sed latinorum in Elenchis apposuit. Nam si ipse Elenchos fecisset, graecorum exempla praetenderet [note that, for this reason, some claim that Aristotle did not write the Elenchi, for Latin examples are offered here rather than Greek ones. Now, if Aristotle had written the Elenchi himself, he would have relied on Greek examples] ».

It goes without saying that the suspicion had no sooner been voiced than the author made short work of it :

[T02] Anonymi glosae in Aristotelis sophisticos elenchos, I, 326.4-8 : « sed dicimus ipsos mentiri, quia Boethius, qui hoc opus de graeco in latino transtulit, exempla latinorum, et non graecorum, dedit, ideo scilicet quia, veluti voces apud latinos et graecos sunt diversae, sic et ipsarum accidentia, id est accentus quibus ipsae voces modulantur [but we maintain that they are wrong, for it was Boethius, who translated the book from Greek into Latin, that used Latin examples instead of Greek ones and he did so because just as Latin words are different from Greek ones, their features are different as well, as in this instance the accents through which words are spoken in verse and measure] ».

The explanation soon became the standard story, as attested time and again :

[T03] Anonymi parisiensis compendium sophisticorum elenchorum, 84.23-28 : « orationum autem in quibus secundum accentum est causa deceptionis duo exempla ab Aristotele ponuntur, unum Horatii, alterum Vergilii. Unde quidam sunt qui coniectant hoc opus non esse Aristotelis, cum illi multo tempore posteriores fuissent illo tempore. Quibus dicendum est quod latinus interpres necessitate coactus est, vel obscuritate graecorum vel difficultate, ponere latina exempla [Aristotle gives two examples of sentences where the deception arises because of the accent : the first example is from Horace, the second from Vergil. For this reason, some have speculated that the work is not by Aristotle, insofar as Horace and Vergil lived long after Aristotle's time. We have to retort that the Latin translator was forced to turn to Latin examples either because of the obscurity of the Greek examples themselves or because of their complexity] ».

[T04] Anonymi Aurelianensis I commentarium in sophisticos elenchos, 123.26-33 : « propter haec exempla non videtur liber iste compositus ab Aristotele, nam primum exemplum ab Horatio, secundum scriptum est a Vergilio, quorum uterque posterior fuit Aristotele. De primo dici potest exemplo quoniam non sumptum est a Vergilio, sed ab Homero quem imitatur Vergilius in opere suo; sed non hoc de exemplo secundo dici potest; dicendum ergo est quod ideo latina ponit exempla quia graeca de verbo translata non idem ostenderent [because of these examples, it seems that the work is not Aristotle's. In fact, the first example is from Horace, whereas Vergil authored the second one and both lived after Aristotle. Concerning Vergil's example, one may argue that it is not his but Homer's, whom he copied in his works. The same cannot be said of the other example ; therefore one has to acknowledge that <the translator> settled for a Latin example because a literal translation of the Greek one would not have been an illustration of the same <fallacy>] ».

[T05] Anonymi SF quaestiones super Sophisticos Elenchos, quaestio 73, 168.20-21 : « exempla quae in hoc libro ponuntur non sunt Aristotelis, sed sunt Vergilii et Horatii, et posita sunt a translatoribus [the examples in the book are not Aristotle's but Vergil's and Horace's. It was the translators that put them there] ».

[T06] Aegidii romani expositio super libros elenchorum, 14rb 5-7 : « exempla hic posita ut communiter dicitur non sunt Aristotelis sed translator ea posuit. Quod ideo fecit, quia exempla forte posita ab Aristotele non errant convenientia nostrae linguae [as it is commonly acknowledged, those examples are not Aristotle's but they were put forward by the translator. And he did so because as they stand Aristotle's examples do not present at all the same problem in our tongue] ».

What of Anonymus C ? what did he have to say about Vergil's and Horace's verses occurring as tokens of fallacies of accent even though -as forcefully pointed out in [T03] and [T04] -neither were around at the time Aristotle wrote the Sophistical Refutations ?

[T07] Anonymi Cantabrigiensis commentarium in Aristotelis sophisticos elenchos, 146.16-24 : « ex hiis exemplis a latinis sumptis volunt quidam convincere hunc librum <non> ab Aristotele graeco compositum esse. Sed forsitan latini nostri graecos imitantes multa dicta a graecis in latinum sermonem transtulerunt, unde non est mirum si in Vergilio et Horatio inveniantur aliqui versus in latinum translati et in graecis †uno † ab Aristotele positi. Vel potest verisimilius dici quod translator huius operis commoda nostrae doctrinae a nostris auctoribus sumpsit exempla, forsitan enim exempla ab Aristotele posita si tranferrentur ambiguitatem non reciperent [because these examples have been borrowed from Latin authors, some want to persuade us that Aristotle did not write this book in Greek. Though, it may be that our Latin authors, who were imitators of the Greek, translated into Latin a lot of what these happened to say. As a result, it is not surprising that one come across in Vergil and Horace some verses used by Aristotle and translated into Latin. Alternatively, and more plausibly, it may be held that the translator of Aristotle's work picked up from our authors a few convenient examples illustrating our matter. In fact, it may well be that, if translated, Aristotle's examples lose their ambiguity] ».

While there is probably no way around the crux « †uno † » in 146.20 -reading « im<m>o » instead of « uno » would not help us much anyway) ; on account of [T03]'s « coniectant », one just might be tempted to favour « connicere » over « convincere » (and accept Sten Ebbesen's alternative emendation instead, that is « conicere »). Be that as it may, the overall meaning of Anonymus C's solution is clear either way : besides stating the obvious (« vel potest verisimilius dici quod etc. »), he got a bit inventive and, along with the right explanation, he came up with the brilliant suggestion ([T04] had done only half the job and got the Vergil Homeric appropriation wrong) that there is nothing wrong to begin with ! Insofar as « Latini nostri » often borrowed from their Greek predecessors, it would not come as a surprise that the disputed examples are the same because the same verses have simply been picked up twice. Se non è vero, è molto ben trovato.

EPILEGOMENA.

A LOOK INTO THE FUTURE OF FALLACIES STUDIES. Bad arguments have never been in short supply. The scholarly interest they have elicited in recent years, on the other hand, is quite exceptional. Fallacies studies have become a well established and flourishing field of argumentation theory ("critical thinking" being one of the umbrella labels under which the subject is often advertised). That being said, with so few exceptions as to make little or no difference, the ever-growing number of papers, book-chapters, book-length studies, reviews special issues and even handbooks on the matter -which are easily counted by the hundreds -suffer from a peculiar lack of interest in Mediaeval theory and practice of argumentation which is -arguably -the most creative stage in the whole history of fallacy theories. The standard story is that after Aristotle got it off to a great start, the discipline became dormant until Richard Whately first and John Stuart Mill soon afterwards revived it in spectacular fashion 5 . This picture, of course, is misleading and deserves to be dismissed or, better still, replaced with a new narrative which brings to bear the full resources of mediaeval treatment of fallacies, possibly across more than one mediaeval tradition. Sten Ebbesen's reliable edition and in-depth study of Anonymus C's commentary on Aristotle's Sophistici elenchi are a giant leap in the right direction. Others will likely follow in his footsteps. Should their work turn out to be even only half as good as his, then we are in for a major turnaround in a field which could certainly use one right now. BIBLIOGRAPHY SOURCES

. A most typical example is 323.11, note

where -as Sten Ebbesen explains in the « Introduction », p. 41 -it is not so much the first honest mistake (a « q a » read as a tironian note, that is as if it were an abbreviation for « contra » rather than for « qua ») that got C into trouble, but the fix to cover the problem up (the accusative form « argumentatione-m » added to match the « contra »). This prompted Sten Ebbesen's emendation « si praesumpserit, i.e. si noverit qua argumentatione debeat uti interrogans » of the transmitted text « si praesumpserit, i.e. si noverit contra argumentationem debeat uti interrogans ».