Jean-David Moreau
email: jean.david.moreau@gmail.com

Vincent Trincal

Louis Baret

Benjamin Bourel

Alain Jacquet

Dinosaur footprints from the Early Jurassic of

Keywords:

During the 20 th century, the first dinosaur tracks of the Causses Basin were identified at Saint-Laurent-de-Trèves, in the Parc National des Cévennes (southern France). A recent excavation reveals a new theropod tracksite in the Hettangian deposits from Le Mazel, 2 km from the historical tracksite at Saint-Laurent-de-Trèves. The tracks are here described combining a biometric approach and 3D imaging photogrammetry. The main track-bearing surface bears 64 in situ tridactyl footprints preserved as concave epireliefs. Two morphotypes were identified, a "Grallatorid" morphotype and a "Kayentapus" morphotype. Footprints belonging to the first morphotype are closely similar to Grallator lescurei, Grallator minusculus, and Grallator sauclierensis. This study shows the difficulty to distinguish quite similar tridactyl tracks from an ichnotaxonomic point of view and highlights the importance of detailed biometric comparisons. Tracks are preserved in a brown to yellowish dolomudstone showing abundant cryptalgal laminites and mud cracks. These deposits were accrued in shallow environments such as intertidal and supratidal zones of a tidal flat.

Mots clés :

Empreintes de pas de théropodes Jurassique Inférieur Hettangien Bassin des Causses Lozère r é s u m é Durant le XX e siècle, les premières empreintes de pas de dinosaures du Bassin des Causses furent identifiées à Saint-Laurent-de-Trèves, dans le Parc National des Cévennes (Sud de la France). Une fouille récente a révélé une nouvelle dalle à traces de théropodes dans les dépôts hettangiens du Mazel, à 2 km du site historique de Saint-Laurent-de-Trèves. Les empreintes sont ici analysées en combinant une approche biométrique avec de l'imagerie 3D par photogrammétrie. La surface principale porte 64 traces tridactyles qui sont conservées in situ sous forme d'épireliefs concaves. Deux morphotypes sont identifiés : « Grallatorid » et « Kayentapus ». Les empreintes de pas du premier montrent de fortes similitudes avec Grallator lescurei, Grallator minusculus et Grallator sauclierensis. Cette étude montre la difficulté à distinguer des traces tridactyles similaires du point de vue ichnotaxonomique et met en évidence l'importance de comparaisons biométriques détaillées. Les empreintes sont imprimées dans une dolomie brune à jaunâtre portant d'abondantes laminites cryptoalgaires et des prismes de dessiccation. Ces dépôts ont été formés dans un environnement peu profond correspondant à un platier tidal.

Introduction

Skeletal remains of Early Jurassic theropods being extremely sparse in Europe [START_REF] Larsonneur | Un dinosaurien carnivore, Halticosaurus, dans le Rhétien d'Airel (Manche)[END_REF][START_REF] Carrano | A review of coelophysoids (Dinosauria: Theropoda) from the Early Jurassic of Europe, with comments on the late history of the Coelophysoidea[END_REF][START_REF] Delsate | The first Early Jurassic (late Hettangian) theropod dinosaur remains from the Grand Duchy of Luxembourg[END_REF][START_REF] Dal Sasso | The oldest ceratosaurian (Dinosauria: Theropoda), from the Lower Jurassic of Italy, sheds light on the evolution of the three-fingered hand of birds[END_REF] and throughout the world [START_REF] Weishampel | The Dinosauria[END_REF][START_REF] Smith | Osteology of Cryolophosaurus ellioti (Dinosauria: Theropoda) from the Early Jurassic of Antarctica and implications for early theropod evolution[END_REF][START_REF] Xing | Tooth loss and alveolar remodeling in Sinosaurus triassicus (Dinosauria: Theropoda) from the Lower Jurassic strata of the Lufeng Basin, China[END_REF], tracksites are important tools to reconstruct dinosaurs from this time period and their palaeoecosystems. Since many Hettangian-Sinemurian archosaur trackways were discovered in the Causses Basin (southern France), the tracksites from this area became capital for the study of Early Jurassic ichnofaunas [START_REF] Ellenberger | La découverte des pistes de dinosauriens de Camprieu[END_REF][START_REF] Demathieu | Problem in discrimination of tridactyl dinosaur footprints, exemplified by the Hettangian trackways, the Causses, France[END_REF]Demathieu andSciau, 1992, 1999;[START_REF] Sciau | Dinosaures et reptiles marins des Causses[END_REF][START_REF] Sciau | Dans les pas des dinosaures des Causses. Inventaire des sites à empreintes[END_REF][START_REF] Sciau | Quelques sites à empreintes de pas de dinosaures connus ou nouveaux[END_REF][START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF][START_REF] Gand | Les traces de pas d'amphibiens, de dinosaures et autres reptiles du Mésozoïque franc ¸ais : inventaire et interpretations[END_REF][START_REF] Moreau | Nouvelles découvertes d'empreintes de dinosaures en Lozère. Analyse biométrique des traces et synthèse paléoenvironnementale de l'Hettangien[END_REF]Moreau et al., 2012a[START_REF] Moreau | Two new dinosaur tracksites from the Hettangian Dolomitic Formation of Lozère, Languedoc-Roussillon, France[END_REF]Moreau et al., , 2018Moreau et al., , 2019aMoreau et al., , 2021a)). Historically, the first dinosaur tracksite mentioned in the Causses Basin was that from Saint-Laurent-de-Trèves, which is located at Cans-et-Cévennes, in the Parc National des Cévennes [START_REF] Monod | Du Musée du Désert au Mont-Aigoual, à l'Aven Armand et aux Gorges du Tarn par la Corniche des Cévennes. Causses et Cévennes[END_REF]Fig. 1). During the first half of the 20 th century, tracks from this locality were interpreted as footprints of a "prehistoric beast" [START_REF] Monod | Du Musée du Désert au Mont-Aigoual, à l'Aven Armand et aux Gorges du Tarn par la Corniche des Cévennes. Causses et Cévennes[END_REF]. It was only during the second half of the 20 th century that they were first analysed by the palaeontologist Louis Thaler (1930Thaler ( -2002) ) from the University of Montpellier (southern France; [START_REF] Thaler | Empreintes de pas de dinosaures dans la dolomie du Lias inférieur des Causses[END_REF]. The Saint-Laurent-de-Trèves tracksite contains over 30 dinosaur footprints and consists of the single tracksite from the Parc National des Cévennes which is equipped for the accessibility to the public (Moreau et al., 2019b).

A recent excavation reveals a new dinosaur tracksite at Le Mazel, 2 km from the historical trackways from Saint-Laurent-de-Trèves (Fig. 1). This new palaeontological locality yields more abundant and better preserved footprints than those from Saint-Laurent-de-Trèves. Here, we describe the tracks using morphometry and 3D imaging photogrammetry, after that the track assemblage is compared with data from other coeval tracks of the Causses Basin.

Geographical and geological setting

The tracksite is located south of the Massif Central (southern France), in the Lozère department, 5 km south of Florac, near the hamlet of Le Mazel (Fig. 1). The exposed deposits at Le Mazel are Hettangian in age. In Lozère, Hettangian deposits are divided into two informal formations [START_REF] Brouder | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Meyrueis[END_REF][START_REF] Briand | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Mende[END_REF][START_REF] Gèze | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Florac[END_REF]: (1) the Detrital Sandstones-Variegated Mudstones Formation and (2) the Dolomitic Formation.

The first formation consists of rubefied lenticular and channelised sandstones that alternate with variegated argillites or marls. Based on pollen grains, this formation was dated from the Rhaetian-Hettangian to Hettangian [START_REF] Grignac | Découverte de microflores d'âge Hettangien dans l'épandage gréso-conglomératique « triasique » formant la base du Mésozoïque de la région de Figeac-Capdenac (Quercy)[END_REF]. Near Florac, the Detrital Sandstones-Variegated Mudstones Formation is poorly developed (a few metres thick) or missing [START_REF] Gèze | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Florac[END_REF]; Fig. 1C) and is largely dominated by white or ferruginous sandstones. In the northern part of the Causses Basin, this formation yields rare fossils. They mainly consist of plant impressions [START_REF] Malafosse | Recherches sur le Lias de la région de Marvejols[END_REF][START_REF] Saporta | Paléontologie franc ¸aise ou description des fossiles de la France, végétaux, plantes jurassiques. Tome IV, types proangiospermes et supplément final[END_REF][START_REF] Moreau | Rediscovery of the allegedly 'destroyed' holotype of Weltrichia fabrei Saporta, 1891 from the Rhaetian?/Hettangian of Lozère (Southern France)[END_REF] ascribed to bennettitaleans (Otozamites latior Saporta, Weltrichia fabrei Saporta emend. Moreau & Thévenard) as well as dinosaur tracks [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF][START_REF] Sciau | Dans les pas des dinosaures des Causses. Inventaire des sites à empreintes[END_REF]Moreau et al., 2012b).

The Dolomitic Formation consists of brown to yellowish dolomite organised in thin or massive layers, as well as dolomitic laminites alternating with green and black argillites or marls that are locally lignitic [START_REF] Gèze | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Florac[END_REF]. The tracksite of Le Mazel is stratigraphically located in this formation (Fig. 1C). Based on rare ammonites (e.g. Psiloceras planorbis Sowerby), the lowermost part of the Dolomitic Formation was dated as Hettangian [START_REF] Brouder | Notice explicative, carte géologique au 1/50.000 de la France, feuille de Meyrueis[END_REF]. In the area of Florac, the formation is 80 to 100 metres thick. Locally, in Lozère, the Dolomitic Formation shows plant-bearing beds yielding conifers (Brachyphyllum paparelii Saporta and Pagiophyllum peregrinum (Lindley & Hutton) A. Schenk emend. Kendall), pteridosperms (Thinnfeldia rhomboidalis Ettingsh) Cycadales (Ctenopteris Brongn. ex Saporta), Filicales (Clathropteris platyphylla Brongn.) and Ginkgoales (Eretmophyllum caussenense Thévenard; [START_REF] Saporta | Paléontologie franc ¸aise ou description des fossiles de la France, végétaux, plantes jurassiques. Tome I, Algues, Equisétacées[END_REF][START_REF] Saporta | Paléontologie franc ¸aise ou description des fossiles de la France, végétaux, plantes jurassiques. Tome III[END_REF][START_REF] Ressouche | Horizon fluvio-lacustre au sommet de l'Hettangien en Lozère[END_REF][START_REF] Roquefort | Contribution à l'étude de l'Infra-Lias et du Lias inférieur des Causses cévenols[END_REF][START_REF] Thévenard | La paléoflore du Jurassique inférieur du bassin des Causses (France). Étude systématique, stratigraphique et paléo-écologique[END_REF][START_REF] Thévenard | Les coniférales du Jurassique inférieur du gisement de Chaldecoste, bassin des Causses (Lozère, France)[END_REF]Moreau et al., 2019cMoreau et al., , 2021b) ) as well as archosaur tracks including dinosaur and crocodylomorph trackways [START_REF] Demathieu | Problem in discrimination of tridactyl dinosaur footprints, exemplified by the Hettangian trackways, the Causses, France[END_REF][START_REF] Sciau | Dans les pas des dinosaures des Causses. Inventaire des sites à empreintes[END_REF][START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF][START_REF] Moreau | Nouvelles découvertes d'empreintes de dinosaures en Lozère. Analyse biométrique des traces et synthèse paléoenvironnementale de l'Hettangien[END_REF]Moreau et al., 2012a[START_REF] Moreau | Two new dinosaur tracksites from the Hettangian Dolomitic Formation of Lozère, Languedoc-Roussillon, France[END_REF]Moreau et al., , 2018Moreau et al., , 2019a)).

Material & methods

The tracksite of Le Mazel was fortuitously discovered by Roger Almies during the 1960s. At that time, he extracted some large blocks of rock for the construction of a fireplace lintel in the small hamlet of Le Mazel. During many decades the location of the tracksites was only known of some residents of the hamlet and a few regional naturalists. In 2012, two of us (JDM and LB) made the first observations after Daniel André and Raymond Lefillatre indicated to us the tracksite location. Only 12 dinosaur tracks were visible on a small outcrop a few metres wide. In July 2018, the Association Paléontologique des Hauts Plateaux du Languedoc (A.P.H.P.L.) organised an excavation (Fig. 2A). Thanks to this excavation, an area of up to 4.5 m wide and up to 22 m long was cleaned, representing a surface of 50 m 2 (Figs. B-C, 3). The main track-bearing surface bears 64 in situ tridactyl footprints preserved as concave epireliefs (Fig. 3).

The descriptive terminology and biometric parameters are taken from [START_REF] Leonardi | Glossary and manual of tetrapod footprint palaeoichnology. Publicacão do Departemento Nacional da Produc ¸ão Mineral Brasil[END_REF], [START_REF] Demathieu | Problem in discrimination of tridactyl dinosaur footprints, exemplified by the Hettangian trackways, the Causses, France[END_REF], [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF] and [START_REF] Marty | Sedimentology, taphonomy, and ichnology of Late Jurassic dinosaur tracks from the Jura carbonate platform (Chevenez-Combe Ronde tracksite, NW Switzerland): insights into the tidal-flat palaeoenvironment and dinosaur diversity, locomotion, and palaeoecology[END_REF]. We used the following standard abbreviations (Fig. 4): "L", length of the footprint; "W", width of the footprint; "LII", "LIII", "LIV" lengths of the digits; "II-IV", divarication angle between digit II and digit IV. We also used "D" which corresponds to the length of the free part of digit III sensu [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]Fig. 4). In the section "Palaeoichnology" below, these morphometric parameters are described by giving three values; the minimum, the average (in brackets) and the maximum. The biometric values of the tracks from Le Mazel were compared with a large dataset including measurements of 427 tridactyl tracks from the Early Jurassic of the Causses Basin (based on data from [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]and Moreau et al., 2012a, 2014, 2018, 2021a). Locomotion speed and height of the hip were estimated based on the formulas of [START_REF] Alexander | Estimates of speeds of dinosaurs[END_REF] and [START_REF] Thulborn | Dinosaur Tracks[END_REF]. In order to produce 3D-photogrammetric reconstructions of the tracksite (e.g. [START_REF] Matthews | Close-range photogrammetry for 3-D ichnology: the basics of photogrammetric ichnology[END_REF], the software Agisoft PhotoScan Professional 1.2.4 was used to align and combine multiple digital photographs taken with a Nikon D5200 camera coupled with an AF-S NIKKOR 18-105 mm f/3.5-5.6G ED camera lens. The same software was used to produce 3D-photogrammetric textured meshes and digital elevation models (DEM) in false colours.

Rock samples were taken from several levels (Fig. 5; Table 1) in order to perform mineralogical analyzes. Thin sections were made for the microscopic observations, powdered for the XRD analyzes. Samples were crushed manually in an agate mortar until a powder with particles < 80 m (sieve control) was obtained. The diffractometer was a Bruker D8 Advance using a copper X-ray tube, equipped with a vertical theta/theta goniometer and a LynxEye XE-TTM high-speed linear detector. A motorised anti-scattering knife was used. The acquisition of the diffractograms was carried out at 40 kV and 40 mA, from 7 • to 70 • 2 with an interval of 0.02 • 2 and a total acquisition time of 2h30 per sample. Minerals were identified using the Bruker-AXS DIFFRACplus Eva v4 software and the 2015 ICDD PDF database. Mineralogical quantification was performed by Rietveld calculation with DIFFRACplus Topas v5 following rules described in [START_REF] Trincal | Quantification of mass transfers and mineralogical transformations in a thrust fault (Monte Perdido thrust unit, southern Pyrenees, Spain)[END_REF].

Results

Sedimentology and track preservation

The excavation conducted in 2018 exposed a 1.7 m high outcrop (Fig. 5). The lithology is quite constant along the stratigraphic section. Exposed beds mainly consist of homogenous centimetre-to decimetre-thick layers of brown to yellowish dolomudstone (94.5 to 98.9 wt% of dolomite and 0.2 to 4.9 wt% of calcite; Fig. 6A,Table 1). It mainly shows a non-porous fabric, however, dolomudstone is sometimes bioturbated (possible root casts; Fig. 6B). Microbial laminites are locally observed (cryptalgal laminites; between 0.4 and 0.7 m; Fig. 5). They are millimetre-thick, parallel to the bedding planes, wavy to planar and show alternating micrite and sparite layers. Mud cracks are abundant (between 0.7 m and 1.1 m; Fig. 6C), whereas ripple marks are less common (only observed at 1.05 m; Fig. 6D).

At the Mazel, five stratigraphic surfaces (S1 to S5) bear dinosaur footprints (Fig. 5). The in-situ footprints exposed on the large trackbearing surface (Fig. 3) are stratigraphically located in S1 to S4. The surface S5 yielded a single track that was extracted during the excavation. Tracks from S1 are covered by a 0 to 2 cm thick lens of dolomite bearing S2. Tracks from S2 are covered by a 0 to 5 cm thick lens of dolomite bearing S3. S3 is overlapped by another 0 to 5 cm thick lens of dolomite bearing S4. The lateral extension of the surfaces S2 and S4 is limited to the northern part of the outcrop. Surfaces S3 and S4 are directly covered by a 1-centimeter-thick layer of clayey dolomite (80.8 wt% of dolomite and almost 14 wt% of phyllosilicates, sample Maz.S6 in Table 1). Since tracks deform a single bed and are not propagated into the underlying layer, they are all interpreted as true tracks. Some of the tracks are exquisitely preserved showing details of pads and claws.

Palaeoichnology

Among the complete traces, two morphotypes were distinguished: a "Grallatorid" morphotype and a "Kayentapus" morphotype. Due to the partial preservation of many footprints, only 33 of the 64 tracks can be confidently assigned to a morphotype.

"Grallatorid" morphotype. The first morphotype is represented by 26 tridactyl II-IV footprints. A single short trackway composed of four footprints is identified (Maz-51-Maz-54); Fig. 7). Pace length is 135-136 cm, and the stride is 271 cm long. Pes tracks are aligned with the trackway midline (Fig. 7). Tracks are small-to medium-sized and mainly longer than wide (L/W = 0.9-(1.2)-1.5), 10.0-(21.1)-29.0 cm long and 10.5-(17.2)-27 cm wide (Fig. 8, Table 2). The angle between II and IV is quite small (T = 29.0-(41.3)-52.0 • ). Digit imprints are well individualised and elongated. Impressions of digits III and II are the longest and the shortest, respectively. The free part of III is quite long (L/D = 1.8-(2.1)-2.8). The position of the digito-metatarsal pad of digit IV is more proximal than for digit II. On some tracks, marks of pads are well preserved and are circular to oval. There is no plantar impression. "Kayentapus" morphotype. The second morphotype consists of 7 footprints. One partial trackway composed of two consecutive tridactyl II-IV tracks (MAZ-1-MAZ-2; Fig. 3) displays a 260 cm pace length. Footprints are as long as wide (L/W = 1.0-(1.0)-1.1), 19.0-(22.9)-28.5 cm long and 18.0-(22.2)-29.5 cm wide (Fig. 8, Table 2). The angle between II and IV is large (T = 44.0-(60.7)-76.0 • ; Table 2). Digit imprints are moderately elongated and wellindividualised. In some tracks, the traces of digits II and IV are quite similar in length. The free part of III is quite long (L/D = 2.0-(2.2)-2.5). The imprints of pads are absent or poorly marked but impression of claws are well-marked.

Discussion

Comparison with tridactyl tracks from the Causses Basin

Six ichnospecies of tridactyl footprints were described from the Lower Jurassic formations of the Causses Basin [START_REF] Demathieu | Problem in discrimination of tridactyl dinosaur footprints, exemplified by the Hettangian trackways, the Causses, France[END_REF][START_REF] Demathieu | Des pistes de dinosaures et de crocodiliens dans les dolomies de l'Hettangien du Causse du Larzac[END_REF][START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF] [START_REF] Welles | Dinosaur footprints from the Kayenta Formation of northern Arizona[END_REF]. The validity of this ichnotaxon has been strongly debated [START_REF] Lucas | Tetrapod footprint biostratigraphy of the Triassic-Jurassic transition in the American southwest. The Triassic-Jurassic Terrestrial Transition New Mexico of[END_REF][START_REF] Lockley | Kayentapus revised: notes on the type material and the importance of this theropod footprint ichnogenus[END_REF]. Some authors consider Dilophosauripus Welles as a junior synonym of Eubrontes (see references in [START_REF] Lucas | Tetrapod footprint biostratigraphy of the Triassic-Jurassic transition in the American southwest. The Triassic-Jurassic Terrestrial Transition New Mexico of[END_REF][START_REF] Lockley | Kayentapus revised: notes on the type material and the importance of this theropod footprint ichnogenus[END_REF]. D. williamsi was broadly used to describe tracks from the Causses Basin (e.g. [START_REF] Demathieu | Des pistes de dinosaures et de crocodiliens dans les dolomies de l'Hettangien du Causse du Larzac[END_REF][START_REF] Sciau | Sur la piste des dinosaures des Causses[END_REF][START_REF] Sciau | Dans les pas des dinosaures des Causses. Inventaire des sites à empreintes[END_REF][START_REF] Demathieu | Empreintes de pas de dinosaures dans les Causses (France)[END_REF][START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF][START_REF] Gand | Les traces de pas d'amphibiens, de dinosaures et autres reptiles du Mésozoïque franc ¸ais : inventaire et interpretations[END_REF][START_REF] Moreau | Two new dinosaur tracksites from the Hettangian Dolomitic Formation of Lozère, Languedoc-Roussillon, France[END_REF]. However, recent morphometrical comparisons made by [START_REF] Gand | Les pistes d'archosauriens : Kayentapus ubacensis nov. isp. (théropodes) et crocodylomorphes du Bathonien des Grands-Causses (France). Conséquences paléo-biologiques, environnementales et géographiques[END_REF] suggested that Dilophosauripus is a synonym of Kayentapus Welles.

Based on the parameters L, T, L/W and (L-D)/D, Fig. 9 shows bivariate diagrams for the diverse ichnospecies from the Lower 2) fall in the range size of G. variabilis (L = 9.0-(11.3)-14.7; [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]. However, they mainly differ from this ichnospecies by a shorter free part of digit III ((L-D)/D of G. variabilis = 0.9-(1.3)-1.9; [START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]. Some of the tracks (e.g. Maz-23; Fig. 8A) are similar to G. sauclierensis, which mainly differs from G. variabilis by smaller L/W values (L/W = 1.3 and 1.5 in average, respectively) and higher II-IV values (51.1 • and 39.4 • in average, respectively). G. sauclierensis was erected by [START_REF] Demathieu | Des pistes de dinosaures et de crocodiliens dans les dolomies de l'Hettangien du Causse du Larzac[END_REF] based on tracks from the Sauclières quarry in Aveyron (Causses Basin, southern France). Most of the largest tracks of the "Grallatorid" morphotype from Le Mazel (e.g. Maz-62; Table 2) clearly fall in the morphometric space of G. lescurei, which was erected by [START_REF] Demathieu | Problem in discrimination of tridactyl dinosaur footprints, exemplified by the Hettangian trackways, the Causses, France[END_REF] based on tracks from Saint-Léons in Aveyron (Causses Basin, southern France). The large size (L = 28 cm) and the small II-IV value (II-IV = 30 • ) of Maz-53 suggest similarities with of G. minusculus (Fig. 8C,Table 2). So, the "Grallatorid" morphotype from Le Mazel is

Depositional environment and trackmakers

The abundance of vertebrate tracks and mud cracks cooccurring with ripple marks shows that the depositional environment was regularly flooded then emerged. Cryptalgal laminites are characteristic of shallow environments such as intertidal and supratidal zones of a tidal flat [START_REF] Matysik | Facies types and depositional environments of a morphologically diverse carbonate platform: a case study from the Muschelkalk (Middle Triassic) of Upper Silesia, southern Poland[END_REF]. At Mende (30 km North from Le Mazel; Fig. 1B) and Marvejols (50 km North-West from Le Mazel), halite pseudomorphs were reported, attesting to evaporitic conditions [START_REF] Moreau | Nouvelles découvertes d'empreintes de dinosaures en Lozère. Analyse biométrique des traces et synthèse paléoenvironnementale de l'Hettangien[END_REF]Moreau et al., 2019b). Such restricted life conditions could explain the absence of invertebrate remains at Le Mazel and their extreme rarity in the Dolomitic Formation.

Based on the osteological architecture deduced from the morphology of the tracks, many authors consider trackmakers of Grallator and Kayentapus as theropod dinosaurs [START_REF] Baird | Triassic reptile footprint faunules from Milford, New Jersey[END_REF][START_REF] Olsen | Type material of the type species of the classic theropod footprint genera Eubrontes, Anchisauripus and Grallator (Early Jurassic, Hartford and Deerfield basins, Connecticut and Massachusetts, U.S.A.)[END_REF][START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF][START_REF] Gand | Les pistes d'archosauriens : Kayentapus ubacensis nov. isp. (théropodes) et crocodylomorphes du Bathonien des Grands-Causses (France). Conséquences paléo-biologiques, environnementales et géographiques[END_REF]. Using the formula of [START_REF] Alexander | Estimates of speeds of dinosaurs[END_REF] and [START_REF] Thulborn | Dinosaur Tracks[END_REF], the hip height of the trackmakers of the "Grallatorid" morphotype from Le Mazel varies from 1.2 to 1.4 m. Their speed varies from 5.9 to 6.7 km/h, suggesting a walking gait (based on the trackway Maz-51-Maz-54). Although more than 60 theropod tracksites were reported from the Hettangian-Sinemurian sediments of the Causses Basin, body remains of theropods are unknown in the Lower Jurassic from this area. In France, only the Hettangian Moon-Airel Formation from Normandie (northwestern France) yielded theropod body remains. They were ascribed to Coelophysoidea, Lophostropheus airelensis Cuny & Galton [START_REF] Larsonneur | Un dinosaurien carnivore, Halticosaurus, dans le Rhétien d'Airel (Manche)[END_REF][START_REF] Cuny | Revision of the Airel theropod dinosaur from the Triassic-Jurassic boundary (Normandy, France)[END_REF][START_REF] Ezcurra | The coelophysoid Lophostropheus airelensis, gen. nov.: a review of the systematics of "Liliensternus" airelensis from the Triassic-Jurassic outcrops of Normandy (France)[END_REF]. Near Marjevols, Moreau et al. (2019a) reported an ichnoassemblage with the co-occurrence of Batrachopus Hitchcock and Grallator attesting the presence of terrestrial crocodylomorphs in littoral ecosystems during the Hettangian.

Based on more than 60 tracksites, the ichnological record of the Dolomitic Formation suggests that the lowermost Jurassic dinosaur communities from the Causses Basin were mainly composed of theropods. They probably crossed the tidal flats in order to hunt along the coastline. Moreover, the tidal flats of the Causses Basin probably corresponded to a way of migration that theropods used to join innermost environments from the Massif Central where they could found nesting and hunting areas.

Wood, pollen and foliar remains collected in the lignitic layers of the Dolomitic formation [START_REF] Barale | La paléoflore jurassique du Jura franc ¸ais : étude systématique, aspects stratigraphiques et paléoécologiques[END_REF][START_REF] Thévenard | Les coniférales du Jurassique inférieur du gisement de Chaldecoste, bassin des Causses (Lozère, France)[END_REF]Moreau et al., 2019cMoreau et al., , 2021b) ) suggest that the Hettangian tidal flats were bordered by littoral forests. Although cycadales, ginkgoales and pteridosperms are present in Hettangian plant assemblage from the Causses Basin, the cuticle remains as well as the pollen strongly suggest that the flora was dominated by conifers (mainly Cheirolepidiaceae; [START_REF] Thévenard | Les coniférales du Jurassique inférieur du gisement de Chaldecoste, bassin des Causses (Lozère, France)[END_REF]Moreau et al., 2021b). [START_REF] Barale | La paléoflore jurassique du Jura franc ¸ais : étude systématique, aspects stratigraphiques et paléoécologiques[END_REF] and [START_REF] Thévenard | La paléoflore du Jurassique inférieur du bassin des Causses (France). Étude systématique, stratigraphique et paléo-écologique[END_REF] estimated that the conifers constituted nearly 98% of cuticles remains. Palaeoecophysiological analyses of cuticles demonstrated that the conifers (Brachyphyllum Brongn., Pagiophyllum Heer emend Harris) from the Dolomitic Formation display xerophytic characters (fleshy shoots, adapted leaf morphology, thick cuticle, and sunken stomata apparatuses limiting the water-loss) which are typical of plants adapted to withstand coastal environments under a tropical climate with contrasted seasons, cyclically dry then humid [START_REF] Thévenard | Les coniférales du Jurassique inférieur du gisement de Chaldecoste, bassin des Causses (Lozère, France)[END_REF]Moreau et al., 2021b).

Fig. 1 .

 1 Fig. 1. A, Location of the Causses Basin in France. B, Causses Basin and location of the study area (black square). C, Simplified geological map of the study area and location of Le Mazel tracksite.

Fig. 2 .

 2 Fig. 2. Illustrations of Le Mazel tracksite. A, Southern part of the tracksite during the excavation. B, Northern part of the tracksite after excavation. C, Southern part of the tracksite after excavation. D, Track-bearing surface (by night) showing deep footprints and mud cracks.

Fig. 3 .

 3 Fig. 3. Le Mazel tracksite and location of footprints. Photographs and interpretative sketches of both southern (top of the figure) and northern (bottom of the figure) parts.

Fig. 4 .

 4 Fig. 4. Morphometric measurements taken on tridactyl tracks. L, length; W, width; D, length of the free part of digit III; LII, LIII and LIV, digit lengths; T, angle between the two external digits.

Fig. 5 .

 5 Fig. 5. Stratigraphic section of Le Mazel tracksite and location of the track-bearing surfaces. "Thi." = thickness (in metres); "Sam." = rock samples; "Lith." = lithology.

Fig. 6 .

 6 Fig. 6. Microfacies and sedimentary structures. A, Dolomudstone. B, Dolomudstone with porous fabric. C, Cryptalgal laminites. D, Mud cracks. Scale bars: A-C = 5 mm, D = 10 cm.

  : Dilophosauripus williamsi Welles, Eubrontes giganteus Hitchcock, Grallator variabilis Lapparent & Montenat, Grallator lescurei Demathieu, Grallator minusculus Hitchcock, and Grallator sauclierensis Demathieu & Sciau. D. williamsi was established based on tracks from the Lower Jurassic of United States of America

Fig. 7 .

 7 Fig. 7. "Grallatorid" morphotype trackway (Maz-52-Maz-54). A, 3D-photogrammetric mesh with texture. B, DEM in false-colour depth map. C, Interpretative sketch. Scale bar = 1 m. Jurassic of the Causses Basin. The two morphotypes from Le Mazel are clearly smaller than the large tracks E. giganteus (L > 30 cm in the diagnosis of Olsen et al., 1998). The "Grallatorid" morphotype from Le Mazel shows a broad range of length values (L = 10.0-(21.1)-29.0) and shares similarities with several Grallator Hitchcock ichnospecies from the Lower Jurassic of the Causses Basin. Some tracks of the "Grallatorid" morphotype (Maz-7, Maz-8, Maz-14, Maz-23, Maz-36 and Maz-42; Table2) fall in the range size of G. variabilis (L = 9.0-(11.3)-14.7;[START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]. However, they mainly differ from this ichnospecies by a shorter free part of digit III ((L-D)/D of G. variabilis = 0.9-(1.3)-1.9;[START_REF] Demathieu | Les traces de pas de dinosaures et autres archosaures du Lias inférieur des Grands Causses[END_REF]. Some of the tracks (e.g. Maz-23; Fig.8A) are similar to

Fig. 8 .

 8 Fig. 8. Tridactyl footprints from Le Mazel tracksite; photographs, DEM in false-colour depth map and interpretative sketch for each track (from left to right). A-E,"Grallatorid" morphotype. F-H, "Kayentapus" morphotype. All scale bars = 10 cm.

Fig. 9 .

 9 Fig. 9. Footprints from Le Mazel tracksite compared to tridactyl tracks from the Early Jurassic of the Causses Basin (based on data from Demathieu et al., 2002; Moreau et al., 2012a, 2014, 2018, 2021a). A, Bivariate diagram L vs T. B, Bivariate diagram L vs. (L-D)/D. C, Bivariate diagram L vs L/W (C). L in metres and T in degrees.

Table 1

 1 Mineralogical composition of Le Mazel samples obtained by XRD analyses (in weight %). Accuracy of the measurement is estimated by multiplying the goodness of fit (GOF) by the standard deviation given by Rietveld software.

	Samples	Calcite	Dolomite	Quartz	Rutile	Illite + Mica	Kaolinite	Rwp	GOF
	Maz.S1	4.9 ± 0.4	94.5 ± 0.5	0.6 ± 0.2	0.0 ± 0.2	-	-	8.96	5.89
	Maz.S2	0.3 ± 0.2	98.1 ± 0.4	1.6 ± 0.2	0.1 ± 0.2	-	-	8.84	5.74
	Maz.S3	1.3 ± 0.3	97.9 ± 0.3	0.8 ± 0.2	0.0 ± 0.1	-	-	8.04	5.30
	Maz.S4	0.2 ± 0.2	98.9 ± 0.3	0.8 ± 0.2	0.1 ± 0.2	-	-	8.03	5.33
	Maz.S6	0.5 ± 0.3	80.8 ± 0.3	4.6 ± 0.4	0.4 ± 0.3	12.3 ± 4.4	1.5 ± 0.7	8.30	5.56
	Maz.S8a	0.9 ± 0.3	97.4 ± 0.5	1.5 ± 0.2	0.2 ± 0.3	-	-	8.98	5.92
	Maz.S8b	1.1 ± 0.3	97.7 ± 0.5	1.1 ± 0.2	0.2 ± 0.2	-	-	8.94	5.87
	Maz.S9	1.3 ± 0.3	97.2 ± 0.4	1.5 ± 0.2	0.0 ± 0.1	-	-	9.05	5.98
	Maz.S10	0.7 ± 0.3	98.5 ± 0.4	0.7 ± 0.2	0.1 ± 0.2	-	-	8.79	5.74

Table 2

 2 Values measured on Hettangian tridactyl tracks from Le Mazel. L: length of the trace; W: width of the trace; D: length of the free part of digit III; LII, LIII, LIV: lengths of the digits; II-IV, divarication angle between digit II and digit IV (all in cm except II-IV in degrees); and ratio L/W, L/D and (L -D)/D.

	Track N •	L	W	D	LII	LII	LIV	II-IV	L/W	L/D	(L-D)/D	Morphotypes
	Maz-1	28.5	29.5	14	16.5	21.5	17	70	1.0	2.0	1.0	"Kayentapus" Morphotype
	Maz-2	27	26	13	8	21	17.5	76	1.0	2.1	1.1	"Kayentapus" Morphotype
	Maz-15	21	20.5	9.5	10	15.5	13	70	1.0	2.2	1.2	"Kayentapus" Morphotype
	Maz-16	19.5	19.5	9.5	9	15.5	13.5	46	1.0	2.1	1.1	"Kayentapus" Morphotype
	Maz-22	19	18	8	9	13.5	11	58	1.1	2.4	1.4	"Kayentapus" Morphotype
	Maz-21	24	22.5	12	12.5	-	16	61	1.1	2.0	1.0	"Kayentapus" Morphotype
	Maz-4	21	19.5	8.5	11.5	13.5	12.5	44	1.1	2.5	1.5	"Kayentapus" Morphotype
	Maz-5	21	17	9.5	-	-	-	49	1.2	2.2	1.2	"Grallatorid" Morphotype
	Maz-7	14	10.5	7.5	5	12	8.5	47	1.3	1.9	0.9	"Grallatorid" Morphotype
	Maz-8	10	11.5	5.5	-	-	-	-	0.9	1.8	0.8	"Grallatorid" Morphotype
	Maz-10	25	18	12	-	19	12		1.4	2.1	1.1	"Grallatorid" Morphotype
	Maz-11	27.5	27	11	16	21.5	17	44	1.0	2.5	1.5	"Grallatorid" Morphotype
	Maz-14	14	10.5	-	-	-	-	-	1.3	-	-	"Grallatorid" Morphotype
	Maz-17	22	18	12	12	18	11	49	1.2	1.8	0.8	"Grallatorid" Morphotype
	Maz-23	14.5	10.5	7.5	6.5	11	9.5	39	1.4	1.9	0.9	"Grallatorid" Morphotype
	Maz-25	26	25	12.5	-	22	15	34	1.0	2.1	1.1	"Grallatorid" Morphotype
	Maz-30	20.5	15	9	-	-	10.5	40	1.4	2.3	1.3	"Grallatorid" Morphotype
	Maz-36	11	10.5	6	3.5	10.5	6.5	45	1.0	1.8	0.8	"Grallatorid" Morphotype
	Maz-37	17	11	8.5	7	14	9	39	1.5	2.0	1.0	"Grallatorid" Morphotype
	Maz-38	21	18	9	-	14	-	36	1.2	2.3	1.3	"Grallatorid" Morphotype
	Maz-39	19	16.5	8.5	8.5	14	9	46	1.2	2.2	1.2	"Grallatorid" Morphotype
	Maz-40	29	27	14	20	25	23	50	1.1	2.1	1.1	"Grallatorid" Morphotype
	Maz-42	15	12	7.5	7	12	8.5	29	1.3	2.0	1.0	"Grallatorid" Morphotype
	Maz-43	22.5	17	12.5	-	19.5	-		1.3	1.8	0.8	"Grallatorid" Morphotype
	Maz-46	24	19	11.5	-	-	12.5	38	1.3	2.1	1.1	"Grallatorid" Morphotype
	Maz-50	-	-	-	-	-	-	-	-	-	-	"Grallatorid" Morphotype
	Maz-51	-	-	-	-	-	-	-	-	-	-	"Grallatorid" Morphotype
	Maz-52	26	-	-	-	26	16	-	-	-	-	"Grallatorid" Morphotype
	Maz-53	28	19	11.5	12	21	18.5	30	1.5	2.4	1.4	"Grallatorid" Morphotype
	Maz-54	25	20	11.5	13	20	14	46	1.3	2.2	1.2	"Grallatorid" Morphotype
	Maz-58	27.5	26	10	14.5	23.5	18	37	1.1	2.8	1.8	"Grallatorid" Morphotype
	Maz-60	20.5	15.5	8.5	11.5	-	12.5	35	1.3	-	1.4	"Grallatorid" Morphotype
	Maz-62	26	21	11	11.5	19	16	52	1.2	2.4	1.4	"Grallatorid" Morphotype

Acknowledgements

We address our sincere thanks to Marcel Almies, the owner of the tracksite of Le Mazel who gave his agreements for palaeontological excavation during the summer 2018. We thank Daniel André and Raymond Lefillatre for information about the tracksite location they gave to us in 2012. We thank Claude Barbini, Guillaume Bessière, Cécile Bridgens, Sarah Laurent, Angéline Nemeth, Louis Pascual and Vivien Pascual for their help during the excavation. We thank the Sebastian Dalman and Spencer G. Lucas for their constructive and thoughtful reviews of the manuscript.

Disclosure of interest

The authors declare that they have no competing interest.


The Martian atmosphere is a very peculiar meteorological and dynamical object. It is essentially composed of CO2, but its average surface pressure, 610 Pa, is less than a percent of the mean terrestrial surface pressure of 1013 hPa. As the Earth, Mars presents a strong seasonal cycle (its obliquity is 26 • ) reinforced by a large orbital eccentricity (e=1.1). The northern summer receives 40% more solar energy than the southern summer, but is much shorter in duration. On both hemispheres, during winter, the atmospheric temperature at the poles decreases below the CO2 condensation temperature: the atmosphere condenses out to form a seasonal polar ice cap, bound to sublime the following summer. This condensation/sublimation cycle induces large, periodic, surface pressure variation with an amplitude of 100 Pa. This cycle is associated with a strong meridional transport of air from the subliming polar cap to the condensing one. An H2O cycle is superimposed on this CO2 cycle: water sublimes from the summer cap, is transported through the equator in Hadley cell circulation, and condenses on the winter cap.

A third cycle, the dust cycle, exists. A variable amount of aerosols, essentially dust, is present in the Martian atmosphere. The dust is lifted by the winds from the dry, desert, solid surface of the planet, and transported, both horizontally and vertically, by the atmospheric circulation. Dusty episodes come in different forms. Dust devils are small vortices, which are present all over the planet. During spring, local or regional dust storms affect both the Northern and Southern hemispheres, but are more intense in the Southern hemisphere. Finally, dust storms can erupt in late southern spring from the Southern hemisphere, and evolve into global, planet-encircling dust storms. The spatial dust activity and the atmospheric loading greatly vary from year to year. For example, global dust storms are not observed each Martian year, but rather with a quasi-period of three years. A good summary of dust storms and their occurrence can be found in [START_REF] Martin | An analysis of the history of dust activity on Mars[END_REF], and GCM simulations aiming to understand the variability and formation mechanisms of dust storms have been presented in Newman et al. (2002a,b). In a thin atmosphere like the one of Mars, the dust optical depth strongly affects the radiative budget and the atmospheric heating rates. For example, when a global dust storms reach an optical depth of one, the atmospheric temperature can rise by as much as 40 K. On the other hand, the boundary layer of Mars turns more neutrally stratified, because the dust also shadows the surface, which does not heat up so dramatically as with a dust-free atmosphere, and convection will not form so easily [START_REF] Määttänen | Sensitivity tests with a one-dimensional boundary-layer Mars model[END_REF].

Global numerical atmospheric models, the so-called General Circulation Models (GCMs) are able to reproduce qualitatively the CO2 and H2O cycles. However, neither GCMs (Newman et al., 2002a,b) nor other atmospheric models (for example, atmospheric boundarylayer and limited-area models), are able to reliably predict the amount of the dust lifted, and especially the interannual variations of the dust activity. Therefore, in order to make quantitative predictions of the meteorological parameters (temperature, surface pressure, winds), in GCMs and other atmospheric models, the dust atmospheric loading needs to be forced to the observed values. This requirement precludes any meteorological forecast for the future, but also hampers the reliability of GCMs to investigate past Martian climates. For all the plotted spectra surface pressure is 6 hPa and the cosine of the incidence angle is 0.2. The albedo is our "reference albedo" and the lines represent different values of τ . The red line is for τ =0.0, the green for τ =0.5, the blue for τ =1.0, the yellow for τ =1.5 and the purple for τ =2.0.

Our project aims at providing new constraints on the Martian dust. We will base our study on data gathered by the spectro-imaging system OMEGA (Observatoire pour la mineralogie, l'eau, la glace et l'activite) aboard the Mars Express spacecraft. The Mars Express mission is operated by the European Space Agency (ESA), while the OMEGA instrument has been designed and built by several laboratories including the LESIA (P.I. Jean-Pierre Bibring, Institut d'astrophysique spatiale, Orsay, France). Since the successful Mars Express insertion into orbit on December 25th, 2003, the instrument has been performing routinely, and has already acquired some 300 million spectra. The OMEGA spectral range includes the visible and near infrared (0.35-5.1 µm), with a spectral resolution of 0.01-0.02 µm. In terms of spatial resolution, the pixel can be as small as 300 meters. OMEGA allows us to retrieve the surface mineralogy, the chemical and physical state of ices, and the atmospheric chemical composition. We will use its ability to determine the atmospheric dust loading, its spatial, seasonal and interannual variations. Our retrievals will be used in association with Martian meteorologists to improve the dust modelling in GCMs and other atmospheric models.

In nadir viewing, it is difficult to distinguish the solar light reflected on dust from the solar light reflected on the surface. But some methods can be used to disentangle surface reflection from dust reflection. At 2.7µm and 4.3 µm, where CO2 absorbs the solar light, no photons can reach the surface. If some light is reflected, it is from dust particles located above 20 km at 2.7 µm and above 70 km at 4.3 µm. This method (see, e.g. [START_REF] Fedorova | Remote sounding of the martian dust from ISO spectroscopy in the 2.7 µm co2 bands[END_REF]) allows us to map the dust optical depth during the whole the Mars Express mission, and will provide insights into spatial and temporal variations of the dust loading. We have developed at the LESIA a radiative transfer code that allows us to treat both the multiple scattering induced by the particles, and the gaseous absorptions. The radiative transfer code can be used in nadir or limb pointing geometry. The synthetic spectra calculated with the code (see examples in Figure 1) are interpolated with respect to known variables as incidence angle, which we get from the OMEGA data itself, and surface pressure, for which we use the Mars Climate Database [START_REF] Lewis | A climate database for Mars[END_REF]. In principle the surface pressure could be extracted from the CO2 2.0 µm band, as described in [START_REF] Haus | Modelling of atmospheric dust extinction and surface reflectance of Mars applying a radiative transfer simulation in the 2.0 µm and 2.7 µm co2 bands[END_REF], and we may try this approach in the next phase. The synthetic spectra need to be interpolated also with respect to surface albedo (not known), which we have evaluated [START_REF] Erard | A spectro-photometric model of Mars in the near-infrared[END_REF] and fit with the interpolation. The last parameter, naturally, is the dust optical depth, which we fit both throughout the whole spectral window used (2.6-3.2 µm) and in the bottom of the absorption band. The two fits are compared, and generally the fit in the bottom of the band gives more reliable results, since the albedo estimation is very difficult and uncertain.

We are mapping the dust atmospheric content during the Mars Express mission using the 2.7µm CO2 band with the technique described above. The results can be compared with mapping done by another Mars Express Instrument (Planetary Fourier Spectrometer, PFS) and survey from the NASA spacecrafts (Mars Odyssey and Mars Global Surveyor). The high spatial resolution provided by the OMEGA instrument will allow us to search for short scale features in the dust cover, such as lee or orographic waves induced by the interaction between the mean wind field and the surface. Such waves, in predictions of the meteorological models, play an important role in the angular momentum transfer between the deep equatorial atmospheric layers to the rest of the planet. Unfortunately, observations of such waves are scarce. The full mapping of these waves provided by OMEGA, would hence be of great interest to meteorological models. A collaboration with the Laboratoire de Meteorologie Dynamique (LMD) will follow the data analysis period to interpret the results within the Martian meteorological frame. We will present analyzed example orbits from different seasons and locations and discuss the observed behaviour of dust.

  CONSTRAINING THE MARTIAN DUST PROPERTIES FROM THE MARS EXPRESS/OMEGA DATA. A. Määttänen, Division of Atmospheric Sciences, University of Helsinki, Finland, T. Fouchet, P. Drossart, R. Melchiorri, T. Encrenaz, M. Combes, Observatoire de Paris, Meudon, France, J.-P. Bibring, Y. Langevin, B. Gondet, F. Poulet, Institut d'astrophysique spatiale, Universite Paris-Sud, Orsay, France, D. V. Titov, Max Planck Institut für Aeronomie, Katlenburg-Lindau, Germany, and the OMEGA team.

Figure 1 :

 1 Figure1: Examples of the synthetic spectra used for fitting. For all the plotted spectra surface pressure is 6 hPa and the cosine of the incidence angle is 0.2. The albedo is our "reference albedo" and the lines represent different values of τ . The red line is for τ =0.0, the green for τ =0.5, the blue for τ =1.0, the yellow for τ =1.5 and the purple for τ =2.0.


Marco Panza

Is Descartes's Geometry an extension of Euclid's?

Introduction

Descartes's Géométrie ( [START_REF] Descartes | La Géométrie[END_REF]), as well as his whole mathematical production, manifest little attention for theorems, and mainly focus on solving problems. This has brought many commentators to describe Descartes's geometry as mainly concerned with problem-solving. Henk Bos has gone so far as to infer from this that Descartes's "primary aim" in doing geometry "was to provide a general method for geometrical problem solving" ( [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], p. 228).

The description is irrefutable. But is it the same for the consequence that Bos (but not only him) has drawn from it? I resist to assent. I argue, instead, that there is a plausible reading of Descartes's Géométrie according to which its main aim is rather that of expounding a genuine mathematical theory, in which the crucial importance of problems and their solution, even their absolute centrality, do not go together with a primary attention for problem-solving per se, that is, with the idea that the ultimate task of geometry is to provide a(n appropriate) solution for as many problems as possible. In other terms, I do not believe that Descartes's attention for solution of problems prefigure a conception of geometry as a collection of methods for solving problems, rather than as a genuine mathematical theory. According to this reading, Descartes's "primary aim" was rather to extend the geometrical theory expounded in the Elements, namely in its first six books (though the fifth concerns, in fact, a tool-the so called theory of proportions-that, despite of being applied in the sixth to the objects studied in the first four ones, also admits a more general application), which I will call, for short, 'Euclid's Plane Geometry'. (As a matter of fact, many of the things I will say here about Euclid's Plane Geometry also apply to solid one. Still, many others either only pertain to the former, or apply to the latter under condition of appropriate adjustments that I would not have room here to detail; also considering that, in the Géométrie, Descartes only deals with plane geometry, I prefer, then, to focus on the former.) The mathematical theory expounded in the Géométrie is just that which results from such an extension. Call it 'Descartes's Geometry'. Promoting it went together with advocating a wide-ranging reform in geometry, and suggesting a fresh foundation of it, both faithful to the spirit of, and deeply innovative with respect to, Euclid's Plane Gemetry: a reform whose import was much larger than making a large class of problems solvable with licensed means.

My present purpose is of shortly accounting for this reform. The main claims I will advance here are better argued for in my [START_REF] Panza | The twofold role of diagrams in Euclid's Plane Geometry[END_REF] and [START_REF] Panza | Rethinking geometrical exactness[END_REF]. Here, I'll come back to them in a less detailed, but, I hope, more incisive and perspicuous way. I also refer the reader to these papers for a number of cases-studies illustrating with appropriate examples the circumstances, distinctions, and options I will mention below.

The role of problems in Euclid's Plane Geometry

The ontology of Euclid's Plane Geometry

In Euclid's Plane Geometry, problems play a crucial and fundamental role: Solving them results in providing this Geometry with its own ontology. Insofar as extending Euclid's Plane Geometry could but have been a matter of extending its ontology, anybody who had aspired to extend it, though remaining faithful to its spirit, should have necessarily looked for new ways for solving problems resulting in the introduction of new (sorts of) geometrical objects. This was just what Descartes did, in my view.

To explain this, I begin with making the following claim: Euclid's Plane Geometry is not about a fixed domain of objects, consistently defined, and supposed or proved to exist once for all; it is rather about some sorts of transient objects available through a licensed construction (where 'available' applies to 'sorts', not to 'objects').

To appreciate the difference, remark that 'to be available' is here intended as a modal verb. By saying that a sort of objects is available through a licensed construction-I will later explain what makes this different from saying both that an object, or some objects are so, and that (all) the objects, or any object of each of the relevant sorts are so-, I do not mean that this construction makes some objects of this sort-or, a fortiori, (all) the objects, or any object of this sort-be brought to existence, or come somehow to be there. I rather mean that objects of this sort are susceptible of being got through such a construction, can result from it. Hence, the first part of the claim is that Euclid's Plane Geometry is about objects, classified as being of different sorts, that could be got through a licensed construction.

Consider equilateral triangles as an example. This part of my claim is that Euclid's Plane Geometry is, among others, about equilateral triangles, insofar as equilateral triangles are susceptible of being got through a licensed construction, that is, are available through a licensed construction, which I take to be essentially different from claiming that (all) the equilateral triangles, or any equilateral triangles are so. As I have just said, I will come back later on this difference. For the time being, it is enough to notice that claiming that equilateral triangles are susceptible of being got through a licensed construction, is another way for claiming that a construction-type is licensed, each instance of which results in an equilateral triangle: a claim whose understanding is perfectly independent of the understanding of what one could mean by speaking of (all) the equilateral triangles, namely of the totality of them.

The second part of my claim pertains to the adjective 'transient'.

What I mean by it is that, once constructed, an object that Euclid's Plane Geometry is about is not kept in stock: it is neither ascribed to a cumulative totality, so as to make it possible, in future, to refer both to it or to the totality itself, nor it is stored for further uses. Keep attention: here I'm speaking of an object, not of a sort of objects. What I just said concerns single objects of a certain sort, not sorts of objects. It concerns, for instance, neither triangles or squares as such, nor triangles or squares so and so, but rather single triangles or squares. What a licensed construction allows to get are single objects of a given sort: a single triangle, or a single square, for example. And what I take to be transient are just these single objects. More than that: typically, once some assumptions are made, relative to the objects which are taken as given at the beginning of the construction, these single objects are determined as such. They are neither arbitrary or generic triangles or squares, nor arbitrary or generic triangles or squares so and so. They are those determined triangles or squares. For example, what the construction expounded in Euclid's solution of Elements I.1 allows to get is just one of the two determined equilateral triangles having the given segment as one of their sides (or possibly both). But then, why did I say that Euclid's Plane Geometry is about some sorts of transient objects? Here comes the third part of my claim. Though a licensed construction cannot but result in single transient objects, what is relevant for Euclid's Plane Geometry is generality. And generality cannot hinge on the construction of single transient objects. It rather hinges on the possibility of permanently constructing single (transient) objects of a certain sort in a licensed way, that is, not on some single objects of that sort (having) being transiently constructed, but on the permanent availability of that sort of objects.

This means that fixing the ontology of Euclid's Plane Geometry depends on answering the following question: Which sorts of objects are available through a licensed construction? That is, which sorts of objects can be got though such a construction? Solving problems just allows answering this question. This is not the only, general, crucial question that doing Euclid's Plane Geometry hinges on answering to. Another is the following: How objects of a certain sort available through a licensed construction are, precisely? That is, how these objects actually are, besides being objects of the relevant sort? For example: how are equilateral triangles, besides being equilateral triangles? How are their internal angles, besides being internal angles of are equilateral triangles?

What allows answering this second question is proving theorems. This is as much as important as answering the first question, or possibly even more, since it is only insofar as this second question is answered that Euclid's Plane Geometry is able to tell something of the objects it is about. Still, this second question cannot be answered within Euclid's Plane Geometry, and, in a sense, can neither be soundly stated, there, if the first is not answered. This makes the first question more fundamental, in a sense. For example, it is only insofar as the sort of equilateral triangles is available through a licensed construction that it is possible to answer, and, in a sense, soundly ask, within Euclid's Plane Geometry any question concerning their internal angles, and then answering by proving, for example, that these angles are all equal to each other.

In spite of the importance of the second question, for my present purpose it is more important to focus on the first one, and on the way it is answered. Still, it is also important to remark that answering neither the first, nor the second question requires the single objects got through a licensed construction be permanent, that is, kept in stock, once they have been constructed. It does not require these objects be ascribed to a cumulative totality, so as to make it possible, in future, to refer both to them, or to the totality itself, since, in Euclid's Plane Geometry, generality does concern no totality of objects, either fixed or cumulative, but rather concerns availability, that is, susceptibility of being got through a licensed construction. It does not require these objects to be stored for further uses, since, if a certain sort of objects is available through a licensed construction, an object of this sort can always be constructed in a licensed way any time this is required.

It follows that, for the solution of a problem to be able to answer the question it is required to answer-and to make then possible to soundly state other questions that proofs of theorems can answer to-, there is no need to ensure any kind of permanent existence to any object. All what solving a problem is required to make is showing that the relevant sort of objects is available through a licensed construction. And it does it by actually constructing a single object of this sort through such a construction. This makes a question arises: how can constructing a single object of a certain sort through a licensed construction provide a proof that this sort of objects is available through such a construction?

What is important to notice, here, is that this question is not concerned with (all) the objects, any object, or the totality of objects of this sort. What is asked here is not how can constructing a single object of a certain sort through a licensed construction provide a proof that (all) the objects, any object, or the totality of objects of this sort are available through such a construction. This latter question would be sensible only if it were possible to give a clear sense to the notion of a totality of objects of the relevant sort. But this is just what is not possible to do within Euclid's Plane Geometry, for the appropriate identity conditions that would make this possible are lacking.

Imagine you are teaching Euclid's Plane Geometry, and you consider an equilateral triangle, represent it by an appropriate diagram on a blackboard, and prove that all its internal angles are equal. Imagine you, then, erase the blackboard, go to another room, face to other students, and make the same: consider an equilateral triangle, represent it by an appropriate diagram on a blackboard, and prove that all its internal angles are equal. Have you dealt twice with the same equilateral triangle, or have you dealt with two distinct triangles? The question is hardly sensible. And being able to answer it is certainly not required for making and teaching Euclid's Plane Geometry. But if there is no way to fix the conditions according to which a question like this can be answered in a way or another, within Euclid's Plane Geometry, there is no more anything as a totality of equilateral triangles that Euclid's Plane Geometry is about.

This having being said, let us come back to the good question: how can constructing a single object of a certain sort through a licensed construction provide a proof that this sort of objects is available through such a construction? This is a question about the universality of proofs in Euclid's Plane Geometry. More in general, the question is this: how can an argument concerned with a single object, o a single configuration of objects, have an universal import? The question is quite difficult, and many answers have been suggested to it (though it has been rarely stated in the appropriate way, in my view). This is not the place for advancing a new answer, or for arguing in favor of one or another of those that have been already offered. All what is important is to notice that the universality of Euclid's Plane Geometry just depends on the possibility of answering to it in a way or another. Here, I merely admit that an appropriate answer is possible, which is the same as recognizing that Euclid's Plane Geometry is endowed with the appropriate sort of universality, though being about some sorts of transient objects, as I have said.

The point that I mainly want to focus on is another. It concerns the notion of a licensed construction, and the role that the identification of the licensed constructions has within Euclid's Plane Geometry. On this matter, my claim is that (appropriately) fixing these constructions complies with a double purpose within this geometry. On one side, it ensures that any such construction is repeatable, with the result that constructing a single object according to such a construction can be taken as a proof that the same can be done in other occasions, which certainly contributes (though in a way that it is not my intention to investigate, here) to make Euclid's Plane Geometry universal. On the other side, it contributes to make Euclid's Plane Geometry a closed theory, that is, a deductive system with sharp limits. In particular, it makes its ontology have sharp limits, though it is not provided by any given totality of objects. This is because Euclid's Plane Geometry is about a certain sort of objects if and only if this sort of objects is available through a construction that is licensed within it. If this is so, we could, then, also say, that this sort of objects is available within this geometry.

All this having been said, we can now simplify our language: provided that a certain sort of objects S is available through a construction licensed within Euclid's Plane Geometry, and, then, within this geometry itself, and t S 's are objects of this sort, I will say that t S 's are in turn available within Euclid's Plane Geometry. Insofar as it does make no sense to speak of (all) the objects, or any object of S (and I will not speak of them, indeed), this does not entail that (all) the t S 's are, or any t S is available within Euclid's Plane Geometry. For example, whereas I will feel free to say that equilateral triangles are available there, I will neither say nor imply that this is for (all) the equilateral triangles or any one of them.

Geometrical Exactness

There should be no need to say that the licensed constructions in Euclid's Plane Geometry are those by rule and compass. What is needed, instead, is to remember that Euclid's never described these constructions this way. This description rather pertains to the way Descartes thought at them, in the framework if his reform. To avoid any risk of explanatory circularity, for the time being, let us call them 'elementary constructions'. They are fixed by a number of clauses, some of which are explicit-namely those stated by postulates I.1-3-, while other are tacit, or only implicitly advanced. This is, for example, the case of the clause that allows a construction to start by taking a certain (finite) number of unrelated segments as given. A construction is licensed if and only if any elementary step of it agrees with one of these clauses.

It follows that extending Euclid's Plane Geometry by extending its ontology, though remaining faithful to its spirit, requires licensing further constructions besides elementary ones. This results, indeed, in making further (sorts of objects) available in geometry, besides those available within Euclid's Plane Geometry. This could be done in different ways: either by relaxing Euclid's clauses is some way, or by jettisoning the 'only if' implication in the previous condition, and locally allowing some constructive step which do not agree with these clauses, or by adding new clauses, or, finally, by replacing Euclid's clauses with a more general precept, entailing them as particular cases.

The first three strategies have all been followed before Descartes, which rather chose the fourth. The question that Bos refers to as that of geometrical exactness in early modern geometry was just that of settling the appropriate strategy to be followed for making new (sorts of) geometrical objects available in geometry besides those available within Euclid's Plane Geometry, so as to make one granted to deal with objects of these sorts, namely either to appeal to them in further constructions and, then, in the solution of problems insolvable within Euclid's Plane Geometry, or to study them as such. Let us try to better grasp the nature of the question.

I begin with an important distinction. Euclid's Plane Geometry could have been extended, and was, indeed, extended, either locally, or globally. By local extension, I mean an extension merely allowing to make available in geometry some particular new (sorts of) objects, with the purpose of appealing to objects of these sorts in solving problems which are (or, at least, appeared to be) insolvable within Euclid's Plane Geometry. By global extension, I mean an extension that not only allows this, but also results in a new organic theory, larger than Euclid's Plane Geometry, but equally closed, in its being, as well, a deductive system with sharp limits (better, a reform which allows the former just insofar as it results in the latter). By saying that Descartes advocated a wide-ranging reform of geometry, and suggested a fresh foundation of it, I mean that he pursued such a global extension of Euclid's Plane Geometry.

What I have said about local extensions could make a puzzle arise: if showing that some (sorts of) objects are available within Euclid's Plane Geometry depends on solving problems, how should one intend the purpose of appealing to new (sorts of) objects, freshly made available in geometry though a local extension of this geometry, for solving problems otherwise insolvable (or apparently so) within it? There is no some kind of circularity here? To see why this is not so, we need to go a little bit farer in inspecting the nature and role of problems in Euclid's Plane Geometry. This is the task of the next sub-section.

Unconditional and Conditional E-objects

The crucial point pertains to the distinction between two quite different ways of specifying a certain sort of geometrical objects (by using the language of Euclid's Plane Geometry, or an appropriate extension of it).

The first way provides an intrinsic specification of the relevant objects considered per se (for the objects specified using the language of Euclid's Plane Geometry, this generally results from a definition explicitly signaled in the Elements, or from a particularization of it). As it is well known, no other objects than points, segments, circles, polygons, and angles-both rectilinear and horn-are available within Euclid's Plane Geometry. Hence, limited to objects available within this geometry, saying that an object is, in this sense, an object of a certain sort results in saying that it is a point, a a circle, a polygon of a certain genus or species, like a triangle, or an equilateral one, a parallelogram, a square, a gnomon, a pentagon, etc., or an anglepossibly an obtuse, an acute or a right one, or a rectilinear or a horn one.

The second way provides an extrinsic specification of objects already specified in the first way, relatively to other such objects. It depends on fixing some appropriate relations that the former objects are required to bear with the latter. Saying that an object is, in this sense, an object of a certain sort results in saying, for example, that it is a point cutting a given segment in extreme and mean ratio, or a square equal to a given triangle, or to a given circle.

For short, I call an object: 'unconditional', insofar as its sort is specified in the former way, and 'conditional', insofar as its sort is specified in the latter way. According to this liberal parlance, a triangle, or also an equilateral or a right-angled one, are unconditional objects, whereas a right-angled triangle having an angle equal to a certain acute given angle is a conditional one, for example. It follows that, for a certain object, to be unconditional or conditional is not to be, as such, so and so, but to be regarded as being so and so. In others terms, the distinction concerns our specification of the relevant objects, not their intrinsic nature.

Let me, now, introduce a further terminological convention: I call 'E-objects' (for 'Euclid's objects') either unconditional points, segments, circles, polygons of any genus and species, and angles-either rectilinear or horn-, or conditional such objects specified, in the language of Euclid's Plane Geometry, relatively to other such objects, only. In short, an E-object is then an object that Euclid's Plane Geometry is concerned with.

By definition, any unconditional E-object can be defined in the lan-guage of this geometry. But some unconditional objects that can be so described are not E-objects. For example, this is so for non-degenerate conics: ellipses, parabolas and hyperbolas. What is more relevant, is not this, however, but rather that not any sort of unconditional Eobjects is available within Euclid's Plane Geometry. For example heptagons are not. Moreover, showing that a sort of unconditional Eobjects is available within Euclid's Plane Geometry can be far from simple. For example, proving this for heptadecagons is quite complex, and was firstly proved only by Gauss ([5] and [START_REF] Gauss | commissis apud G. Fleischer Jun[END_REF], Sect. VII). A first crucial purpose of solving problems in Euclid's Plane Geometry is, then, to prove that a certain sort of unconditional E-objects is available within it. The solution of Proposition I.1 shows this for equilateral triangles, for example.

For conditional E-objects the dyscrasia is even more evident. Any sort of conditional E-object can be defined in the language of this geometry, but many sorts of conditional E-objects are not available within it. For example, pairs of segments providing two mean proportionals between two given segments, angles trisecting a given rectilinear angle, or squares equal to a given circle, are not. Another crucial purpose of solving problems in Euclid's Plane Geometry is to prove that a certain sort of conditional E-objects is available within Euclid's Plane Geometry, which means that single objects of this sort can be got through an elementary construction, starting from the objects relatively to which they are specified. The solution of Proposition I.2 shows this, for example, for segments equal to given ones, and having one of their extremities in a given point.

It should now not be difficult to see why what I have said above generates no circularity. One could appeal to some (sort of) unconditional objects (either E-ones or not), freshly made available in geometry though an extension of Euclid's Plane Geometry, for solving problems otherwise insolvable (or apparently so) within this geometry, and so making some (sorts of) conditional objects (typically E-ones), which are not available within it, become available in geometry. For example, Menaechmus showed how to appeal to appropriate conics in order to construct a pair of segments providing two mean proportionals between two given segments ([1], vol.III, pp. 92-97; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 38-40), which suggested to locally extend this geometry by licensing the construction of conics.

Extending Euclid's Plane Geometry Before Descartes

Tough quite broad and general, what I have said up to now should be enough, considering the quite limited purpose of the present paper, for fixing those aspects of Euclid's Plane Geometry that are relevant, in my view, for understanding the nature of Descartes's reform. I can, then, stop here concerning this matter, and coming to another matter, also useful for understanding the nature of this reform, namely to the state that the question of geometrical exactness had reached before Descartes.

Euclid's Plane Geometry is far for exhausting pre-Cartesian, and even Greek geometry. Still, with the only possible, partial exception of Apollonius's theory of conics, no other portion of pre-Cartesian geometry looks like it, that is, as a closed organic theory. Before Descartes, and with this only partial exception, going beyond Euclid's Plane Geometry was the same as entering a much more slipping area.

Moving in this area required handling non-elementary constructions: on the one side, these were constructions of E-objects non available within Euclid's Plane Geometry, especially of conditional ones, like segments providing two mean proportionals between two given segments, angles trisecting given angles, or squares equal to given circles; on the other side, these were constructions of geometrical objects other that E-ones, typically curves other than circles.

The most part of these curves were defined by describing possible constructions of them. Admitting these constructions came then, ipso facto, together with warranting for their availability in geometry. Hence, concerning them, the relevant question was less that of looking for appropriate constructions, than that of arguing in favor or against the admissibility of the constructions of which they were described as being the outcomes. A natural way for arguing in favor of it was by emphasizing that admitting these constructions (and taking, then, the sorts of curves constructed by them to be available in geometry) al-lowed to construct (in a suitable way) E-objects non available within Euclid's Plane Geometry. These made the question of the admissibility of (non-elementary) constructions geometrical objects other that Eones an aspect of the question of the admissibility of (non-elementary) constructions of E-objects non available within Euclid's Plane Geometry.

Asking for the construction of an E-object object non available within Euclid's Plane Geometry was to state a problem, namely one that I suggest to call a 'quasi-Euclidean problem'. For its own nature, such a problem required a solution outside Euclid's Plane Geometry, that is, a solution depending on a non-elementary construction. Understanding Descartes's reform of this geometry is manly helped by looking at the different ways suggested, before him, for solving quasi-Euclidean problems.

I have no space here for parsing examples. I confine myself to say that the angle-trisecting, the two-mean proportionals, and the circlesquaring problems were among the most relevant and usual quasi-Euclidean problems that mathematicians dealt with, before Descartes. Plenty of solutions of these problems (and other similar) were offered, differing to each other not only in their details, but also, and overall, for the general strategy they were following and the very nature of the non-elementary constructions involved in them. I suggest distinguishing six different families of non-elementary constructions that one can found in these solutions.

Firstly, come two families of what I call 'quasi-elementary constructions': constructions relying only on E-objects, though applying constructive clauses non included among those of elementary constructions.

The first -family includes constructions using some instruments for getting some points under the condition that some elements of these instruments coincide with (appropriate elements of the diagrams representing) some given geometrical objects. In my parlance, these are constructions using mechanical instruments in the pointing way. An example is provided by Eratosthenes's solution to the two-mean proportionals problem related by Pappus ([12], vol. I, pp. 56-59; [START_REF] Knorr | The Anctient Tradition of Geometric Problems[END_REF], p. 211; [START_REF] Knorr | Textual Studies in Ancient and Medieval Geometry[END_REF], pp. 64-65), involving three rectangular plates, sliding to each other up to assume a position in which a point on one of them falls on the right line passing through a vertex of another and the intersection point of a side of the latter and the diagonal of the third.

The second family includes constructions appealing to some explicit stipulations or tacit admissions working as new constructive clauses. Viète's neusis postulate ( [START_REF] Viète | Supplementum geometriae. Jamet Mettayer[END_REF], prop IX; [START_REF] Viète | The analytic art. Nine studies in algebra, geometry and trigonometry from the Opus Restituitae Mathematicae Analyseos, seu algebrâ novâ[END_REF], p. 398; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 167-173) provides an example.

Then come four families of what I call 'strictly non-elementary constructions': constructions relying on some objects other than Eobjects, namely same curves other than circles. They differ to each other for the way these curves are constructed.

The first family includes constructions involving conics which were supposed to be ipso facto constructed for their being univocally determined. Menaechmus's solution of the two-mean proportionals problem mentioned above provides an example. Another is provided by a solution of the angle-trisecting problem offered by Pappus ([12], vol. I, pp. 271-277; [START_REF] Knorr | Textual Studies in Ancient and Medieval Geometry[END_REF], pp. 213-216; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 53-56), in which a neusis is not merely granted according to a suitable postulate (as later suggested by Viète), but rather constructed by intersection of a circle and a hyperbola.

The second family includes constructions involving curves traced by instruments used to trace them, that is, as I say, in the tracing way. An example is provided by the solution of the two-mean proportionals problem involving a conchoid traced through an appropriate instrument, as suggested by Nicomedes and related by Eutocius ( [START_REF] Archimedes | Archimedis Opera Omnia, iterum editit[END_REF], vol. III, pp. 114-127; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 3133), or by any constructions involving conics that are supposed to be traced by a "perfect compass" ( [START_REF] Woepcke | Troits traités arabes sur le compas parfait[END_REF]).

The third family includes constructions involving loci that were supposed to have been constructed if a generic point of them had been so, starting from the supposition that their generating point is given. An example is provided by the solution of the two-mean proportionals related by Villalpando ( [START_REF] Prado | In Ezechielem explanationes et apparatus urbis ac templi hierosolymitani. Commentariis et imaginibus illustratus[END_REF], vol. III, pp. 289-290; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 75-78).

Finally, the fourth family includes constructions involving curves resulting by interpolation from countably many points previously constructible through an indefinitely protractible (licensed) procedure. An example is provided by the solution of the circle-squaring problem in-volving a quadratrix constructed by points, according to Clavius's suggestion ( [START_REF] Clavius | Elementorum Libri XV accessit XVI de solidorum regolarium cuiuslibet intra quodlibet comparatione[END_REF], vol I, pp. 895-896; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 160-166).

Descartes's Reform

A rapid way to account for Descartes's reform is by accounting for his attitude with respect to non-elementary constructions.

The first thing to be said is that his aim was not merely to accept some of them while rejecting the others, by relying on some good reasons. It was rather to integrate some of them within a new theory extending Euclid's Plane Geometry, and being as close and organic as it. In other terms, what he was looking for was not some scattered sorts of admissible non-elementary constructions, but a global extension of this geometry grounded on a unique principle to be used to license a unique sort of constructions, including elementary ones, but going largely beyond them, so as to allow the solution of the most part of quasi-Euclidean problems.

I suggest referring to this unique sort of construction as to constructions by rule, compass and reiteration. The reason is that they obey the following idea: one begins by admitting that whatever is constructible by an elementary construction is also constructible by such a construction; one can then appeal to an elementary construction to construct appropriate configurations of E-objects available within Euclid's Plane Geometry, such that conceiving of them as movable makes a point of them describe a curve; any curve so described is also taken to be constructible by such a construction; by appealing to whatever such curves and acting on them according to the clauses of an elementary construction (in which these curves are taken as given), one can then construct new configurations of objects (including these very curves), which are also such that conceiving them as movable makes a point of them describe a new curve, which is, in turn, taken to be constructible by such a construction; one can then continue this way indefinitely, by constructing new and new sorts of curves, and new and new configurations of objects (including these new curves), all such that conceiving them as movable makes a point of them describe a further curve, which is both taken as constructible by such a construction, and possibly used, in the same way as the previous ones, to construct other curves.

It is clear that the movable configurations entering such a construction are supposed to play the role that Descartes ascribes to his tracing instruments. What I argue for is, then, that these instruments have to be regarded as bona fide geometric objects, though movable ones; namely objects available within the extended geometry, i. e. Descartes's Geometry, in same sense as E-objects constructible by an elementary construction are available within Euclid's Plane Geometry: they are both so for they are susceptible of being got through a licensed construction, in Descartes's and within Euclid's Plane Geometry, respectively.

Still, for it to be appropriate for entering a construction by rule, compass and reiteration, a movable configuration has to be so determined by this very construction that it be enough to chose a point on it and to identify its movable and fixed -parts (with respect to the surface on which the curve is to be described), for entirely determining a unique curve that it traces through its motion. Of course, one can suppose that such a configuration is moved by some external mover, for example by a human hand, but the way this mover acts on it, and, in particular, the speed it imparts to it, have to have no influence on the trajectory of its movable parts, and, then, on the curve that it describes by its motion. This mover is then to be supposed to act on a unique (movable) point of the configuration, and whatever this point might be, and whatever the mover acts on it, the curve traced by the configuration is to be the same. Moreover, the transmission of the motion from the point on which the mover acts to the other movable points of the configuration -and, in general, from any its movable point to any other-has to merely depend on the requirement that the conditions fixed by construction be conserved through the motion, and to be, in particular, independent of any internal force or mechanical constraint. To say it shortly, a movable configuration entering a construction by rule, compass and reiteration is required to be a purely kinematical devise (having only one degree of freedom), not a dynamical one.

Consider a circle wheeling on a straight line without sliding on it.

A fixed point of it describes a curve, namely a cycloid, which is entirely determined by the circle itself and the mere admission that it wheels on a straight line without sliding on it. Still, this is not a movable configuration appropriate for entering a construction by rule, compass and reiteration, and cycloids are not curves constructible by such a construction, since a circle that wheels on a straight line without sliding on it is a dynamic devise, not a pure kinematical one: for it to move this way, its rotation has to be coupled with a rectilinear motion of its center, which, if it not independently imparted to it, is to induced by some friction. A similar argument also applies to tracing instruments involving strings and pulleys, if the way they work depends on the fact that their strings and pulleys transmit motion by exerting a force, as in the case of Huygens's instrument for tracing spirals: [START_REF] Huygens | OEuvres complètes de Christiaan Huygens[END_REF], vol. 11, p. 216; [START_REF] Bos | Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction[END_REF], pp. 347-349). Imposing these conditions on movable configurations entering a construction by rule, compass and reiteration is quite natural, if one wants these configurations be regarded as genuinely geometric devises. These conditions are, moreover, just those that are met by a tracing rule (a fixed straight line with a tracing point moving on it) and a usual compass (a segment rotating around one of its extremities, whose other extremity is a tracing point). This makes these configurations natural extensions of the tracing devises involved in elementary constructions, conceived as constructions by rule and compass. In my view, this perfectly explains why Descartes call 'geometrical' the curves traced by them and 'mechanical' those that cannot be so traced (like the cycloid), and insists that the former have to be admitted in geometry, while the latter have to be excluded by it. Though this is not Descartes's use, it is then also natural to call, by extension, 'geometrical linkage' a movable configuration involved in a construction by rule, compass and reiteration.

Just like E-objects, geometrical linkages, and, more generally, any objects constructible by rule, compass and reiteration, can be either unconditional or conditional, according whether they are specified per se, or in relation to other given objects. By definition, both geometrical linkages and, more generally, objects constructed by rule, compass and reiteration are available within Descartes's Geometry. Un-conditional ones are so insofar as they are constructible by rule, compass and reiteration starting from any number of unrelated segments taken as given (just as for E-objects within Euclid's Plane Geometry). Conditional ones are so insofar as they are constructible by rule, compass and reiteration starting by the objects relatively to which they are specified, which have to be, in turn, available within Descartes's Geometry. A conditional geometrical linkage is supposed to enter the construction (by rule, compass and reiteration) of a conditional geometric curve (other than a circle), if this it is required to in a particular position (relatively to given objects), which can, in turn, be required in order to solve a quasi-Euclidean problem or to construct another conditional curve by rule, compass and reiteration.

Many other details and specifications could be given (an should be so, in a more comprehensive exposition), and many examples offered. But the little I have said should be enough to make clear how licensing constructions by ruler, compass and reiteration extends Euclid's Plane Geometry. The only further point that I want to make concerns Descartes's attitude with respect to the six families of non-elementary constructions considered above. This attitude comes naturally together with the choice of globally extending Euclid's Plane Geometry by admitting, besides elementary constructions, also (and only) constructions by rule, compass and reiteration. In my view, the latter is, even, a natural outcome of the former, and can then be explained by it.

It seems quite clear, firstly, that Descartes rejected, from the very beginning, quasi-elementary constructions. He possibly rejected those based on specific stipulations, like Viète's neusis postulate, for he considered these stipulations ad hoc, and those involving instruments used in the pointing way, for this way of using instruments is not only essentially different from the way rules and compasses can be taken to be used in elementary constructions, but also since it induces a use of diagrams which is, also, essentially different from the use they have in Euclid's Plane Geometry. This last point deserves an explanation. The difference pertains to this: using instruments in the pointing way allows performing the relevant constructions only if these instruments are conceived as physical objects that, by an appropriate (typical visual) inspection, are recognized to take some required positions with re-spect to the relevant diagrams; rules an compasses can, instead, to be taken to enter an elementary construction at most insofar as they are used as devises for tracing (diagrams representing) segments and circles in positions that they are supposed to take (without any need of verifying that they do, through any sort of inspection, either visual or of any other kind). The latter is, by the way, also what happens, mutatis mutandis, for geometrical linkages in constructions by rule, compass, and reiteration.

Concerning strictly non-elementary constructions, Descartes seems to have, mutatis mutandis, the same attitude toward all of them: taking them as licensed if and only if that can be regarded as constructions by rule, compass and reiteration. Since a conic univocally determined relatively to given objects can always be traced through an appropriate geometrical linkage, he straightforwardly admitted those involving conics taken as ipso facto given for their being univocally determined. He admitted, then, those involving curves traced by instruments used in the traced way, if and only if these instruments can be re-conceived as (or replaced by) geometrical linkages, and those involving loci that are supposed to have been constructed if a generic point of them has been so, if and only if these loci can be traced by such a linkage (whose construction is often directly suggested by the very construction of this generic point). Finally, insofar as, in general, the identification of an indefinitely protractible (licensed) procedure apt to construct countably many points provides no suggestion for constructing a geometrical linkage tracing a curve interpolating these points, Descartes rejected those strictly non-elementary constructions that involve such a curve, unless it were not independently shown how to construct this curves through a construction by ruler, compass and reiteration.

To complete the account of Descartes's reform, one should of course, besides adding a lot of details, explaining how he suggested to associate geometrical curves with polynomial equations, and to chose, among different solutions of the same problem, all depending on a construction by ruler, compass and reiteration, the one regarded as the simplest one, according to an algebraic criterion of simplicity pertaining to these equations. This is not something I can do here, however. My aim was only to show how Descartes's Geometry directly stems from extending Euclid's Plane one. The importance of his associating geometrical curves with polynomial equations can hardly be overestimated, also from the point of view from which I have suggested to look at the former geometry, since it suggested the path that mathematicians followed over the further two centuries, and that finally brought to mathematical theories essentially different from the latter, insofar as they are based on a structurally different ontology. In my view, the fact remains, however, that the introduction polynomial equations for expressing geometrical curves was a posterior step in the fashioning of Descartes's Geometry, a step which was made possible, among other things, by the way this geometry stemmed from extending Euclid's Plane one.