Perrine Pittet
email: ppittet@articque.com

Jérôme Barthélémy
email: jbarthelemy@articque.com

An Application Ontology to Help Users of a Geo-decision Software Understanding their Data

Keywords: OWL DL, Description logics, application ontology, ontology development, semantic annotation, Cartes & Données, geo business software

This paper intends to describe the application ontology of the SaaS version of the decision statistical mapping and geomarketing software Cartes & Données (C & D): CD7 Online. Specified in OWL DL, the CD7 ontology was conceived for automation of semantic annotation of CD7 Online user data to help users better understand their data and make better selection and representation choices when building maps.

Introduction

Ontologies have been introduced in the Semantic Web research field in the early 2000's to exploit textual documents available on the Web in formalized information [START_REF] Berners-Lee | Semantic Web Stack[END_REF]. As such, they are sometimes presented as tools for knowledge representation adapted to the Web environment, to automatically transform data into information and information into knowledge [START_REF] Kaladzavi | OntoSOC: Sociocultural Knowledge Ontology[END_REF]. In this paper, we describe an ontology, which was developed to foster users understanding regarding their data within a geo business decision SaaS application called CD7 Online 1 . This ontology, specified in OWL DL, supports the automatized semantic annotation process of user data. In our case the annotation process generates a graph of RDF annotations for each user data workspace, which is stored as a namedgraph in a triplestore. Each namedgraph is automatically queried by an interactive visualization tool, on which users navigate to discover the knowledge behind their data. The rest of the paper is articulated in 3 sections. The second section presents the CD7 Online project background to expose our motivations for developing a formal application ontology and how this ontology can help users better understand their data. The third section describes the CD7 ontology main concepts and justifies their use regarding the task of semantic annotation of user data, which is supported by the ontology. The third section shows the results of the integration of the ontology within the CD7 Online software.

Project Background

CD7 Online is the SaaS application of the 7 th version of Cartes & Données2 (C & D), which is a French commercial decision statistical mapping and geomarketing software, published by Articque3 . C & D allows users to obtain effective and interoperable maps built on statistical data, without being mapping specialists. As a business decision tool, it is a data analysis and visualization oriented application, which aims at helping people to take decisions via the maps they build upon geovisualization. C & D has been designed since the very beginning with the aim of being self-explanatory, simple, and highly intuitive for users -ease-of-use being a major requirement. Nevertheless, C & D still relies on the users good knowledge of their data and their ability to choose the relevant analysis and representation tools to build meaningful maps. But most of C & D users have a punctual use of the software and often do not have enough available time to study and fully exploit the potential of their data. For solving these issues in CD7 Online, we decided to provide users the knowledge they require to quickly understand their data and their potential applications. We chose to use an automated semantic annotation process on these data in order to extract and represent this knowledge. Automatized semantic annotation of data is the process of automatically associating relevant metadata to data, so that each data is described by a set of semantic annotations. The main objective is to exploit these annotations to allow users visualizing, via an interactive graph navigation tool, the concepts related to their data and the semantic relations they share. This tool allows them to intuitively navigate in the annotations, compare and select relevant data to build relevant maps (cf. Fig. 1). As CD7 Online user data consist in statistical and geographical data tables, we adapted a methodology suited to semantic annotation of tables of data proposed in [START_REF] Hignette | Annotation sémantique floue de tableaux guidée par une ontologie[END_REF]. In [START_REF] Hignette | Annotation sémantique floue de tableaux guidée par une ontologie[END_REF], an ontology of the food microbiology domain is adapted to support a semantic annotation process. The concepts of this ontology, cover the definitions of microbiological symbolic and numerical types, units, value intervals, relations shared by types and the corresponding lexical data, which are used to name them. We similarly developed an ontology describing the knowledge underlying the geographical and statistical data used by CD7 Online. Also, because this knowledge strongly depends on the CD7 Online application specific uses and processes, this ontology is not a domain ontology as in [START_REF] Hignette | Annotation sémantique floue de tableaux guidée par une ontologie[END_REF]'s methodology but an application ontology [START_REF] Malone | Reference and application ontologies[END_REF]. This however does not alter the efficiency of the semantic annotation process. The methodology was actually designed to accept any ontology in which semantic relations with lexical data can be added, in order to make possible lexical similarity measures. For the development of the ontology we follow a simple methodology proposed in [START_REF] Noy | Ontology development 101: A guide to creating your first ontology[END_REF]. The ontology development and evaluation experience was presented in [START_REF] Pittet | Experience of Formal Application Ontology Development to Enhance User Understanding in a Geo Business Intelligence SaaS Platform[END_REF]. The following section focuses on the description of the main concepts of the ontology.

CD7 Application Ontology description

For the purpose of this article, we rely on the ontology definition of [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. Therefore, we define the application ontology as a formal explicit description of concepts of the CD7 Online application data domain, properties of each concept describing various features and attributes of the concepts, and restrictions on properties. The ontology together with the set of individual instances constitutes the knowledge base designed for the automatized semantic annotation process. A concept can have subconcepts representing concepts that are more specific than this concept. Properties describe properties of concepts and instances. As we needed to keep the maximum expressiveness while retaining computational completeness, decidability for inference purposes, we chose to specify the ontology in the OWL DL language. Note that the CD7 ontology terms are originally written in French. Somehow, to facilitate the reading of its description, we translated terms in English and use description logic [START_REF] Baader | Basic description logics[END_REF] in the following part.

The CD7 ontology 4 defines two main concepts: DataComponent and CDComponent. DataComponent describes all components related to user data, such as metadata, user data. CDComponent describes all components related to the CD7 Online specific application processes applicable to user data. All the other concepts fall under these two concepts. Due to lack of space we will focus on the main DataComponent underlying concepts, which are dedicated to semantic annotation. Three main concepts are considered here: UserData, Metadata and LexicalData.

UserData is a DataComponent englobing the three types of data files a CD7 Online user can have in a group of his workspace and use within CD7 Online, such as statistical data files, basemaps and maps. UserData is defined in SHOIN(D) DL axioms as follows:

UserData ⊑ DataComponent ⊓ 1 hasFilename.FileName ⊓ ≥1 ownedBy.User ⊓ 1 hasGroup.Group UserData ≡ StatisticalDataFile ⊔ Basemap ⊔ Map
• with StatisticalDataFile designating the statistical data files, such as Excel files, which contain at least one data table defined by:

StatisticalDataFile ⊑ UserData ⊓ ≥1 hasDataTable.DataTable
• with Basemap designating basemap files used in maps, containing geographical data of a certain geographical space at a certain geographical level, defined by:

Basemap ⊑ UserData ⊓ 1 hasGeographicalSpace.GeographicalSpace ⊓ 1 hasGeographicalLevel.GeographicalLevel
• with Map designating the map project files created by users within CD7 Online, which can import basemaps and statistical data columns, defined by:

Map ⊑ UserData ⊓ ≥0 hasBasemap.Basemap ⊓ ≥0 hasDataColumn.DataColumn
Metadata is a DataComponent designating all the metadata concepts that can be used to describe the underlying knowledge of components of user data or user data themselves in semantic annotations. Metadata is defined by:

Metadata ≡ DataType ⊔ GeographicalLevel ⊔ GeographicalSpace ⊔ DataIndicator ⊔ Theme ⊔ Date ⊔ Unit ⊔ WeightedTerm ⊔ WeightedWord
• with DataType covering the three types of data types that can qualify a data column in a data table of a statistical data file: quantitative data, qualitative data and discrete data.

DataType ≡ QuantitativeData ⊔ QualitativeData ⊔ DiscreteData ⊔ IdData ⊔ UnknownDataType DataType ⊑ Metadata ⊓ ≥0 hasDataType -.DataColumn
• with GeographicalLevel defining all the geographical division levels that can be considered in a statistical data file or a basemap (ex: regional, national level, etc.).

GeographicalLevel ⊑ Metadata ⊓ ≥0 hasGeographicalLevel - .DataTable
• with DataIndicator describing all the statistical indicators a data column can be related to (ex: GDP, mortality rate, etc.). Statistical indicators are categorized by themes. Each statistical indicator is associated with at least one weighted term representing the potential composition of weighted lemmas generally used to designate this indicator.

DataIndicator ⊑ Metadata ⊓ ≥0 hasDataIndicator -.DataColumn ⊓ ≥1 hasTheme.Theme ⊓ ≥1 hasWeightedTerm.WeightedTerm
Another sort of DataComponent is used to support the lexical similarity measures used to determine the statistical indicator related to a data column: LexicalData. LexicalData designates two lexicons instantiated from two concepts: WeightedTerm and WeightedWord.

• with WeightedTerm defining a lexicon of all the instances of weighted terms related to statistical indicators that can be compared with data column title lemmas through lexical similarity measures in order to determine the statistical indicator related to the data column. A weighted term is composed of weighted words.

WeightedTerm ⊑ DataComponent ⊓ ≥1 hasWeightedWord.WeightedWord

• with WeightedWord defining a lexicon of all the instances of weighted words that can compose weighted terms. A weighted word is described by a text and a weight, which are respectively typed with string and float values. Additionally, a set of properties, representing the relations between user data components and metadata (as illustrated above) has been defined. Their domains, ranges and facets have also been formalized (c.f. [START_REF] Pittet | Experience of Formal Application Ontology Development to Enhance User Understanding in a Geo Business Intelligence SaaS Platform[END_REF]). Finally, in order to set up the knowledge base for the semantic annotation task, a set of individuals was instantiated from the concepts WeightedWord, WeightedTerm, DataIndicator, Datatype, Theme, Unit, GeographicalLevel, GeographicalSpace. These individuals were required for the automatized semantic annotation process. For example, to identify and annotate a data column with a statistical indicator, the process evaluates the lexical similarity of data cells content with instances of WeightedWord, which are composed of instances of WeightedTerm associated to instances of DataIndicator. Below is illustrated an example of such instantiation:

WeightedWord(w_mortality0 .

Conclusions and future works

The CD7 application ontology is part of the CD7 Online project semantic layer development and supports an automated semantic annotation tool. This tool produces annotations of user data browsable through a graph navigation tool that users can use to better understand their data and build better maps. CD7 Online being a commercial software, the development and integration of this layer follows its successive updates. Until now it involved the integration of many semantic tools and technologies. Adding such amount of new technologies in an industrial project where deadlines strongly matter was a challenge. Hopefully using W3C standards such as OWL clearly helped to reduce development time as many compatible tools for edition, deployment, querying, management and evaluation exist: Protégé, Pellet, Apache Jena-Fuseki, SPARQL, etc. Today the CD7 ontology and semantic layer still evolves to stick to the software new features. We are working on adding SWRL rules to provide CD7 Online users suggestions of statistical and geographical analysis processes and map representations within a recommender system.

Figure 1 :

 1 Figure 1: Visualization of user data annotations within the CD7 graph navigation tool.

WeightedWord⊑

  DataComponent ⊓ ≥0 hasWeightedWord -.WeightedTerm ⊓ 1 text.xsd:String ⊓ 1 weight.xsd:float

CD7 Online: https://cdonline.articque.com/

C & D website: http://www.articque.com/solutions/cartes-et-donnees/

Articque website: http://www.articque.com/

CD7 Ontology url: http://support-articque.com/ressources/CD7Ontology.owl


Perrine Pittetand

Jérôme Barthélémy
email: jbarthelemy@articque.com

Exploiting Users' Feedbacks: Towards a Task-based Evaluation of Application Ontologies throughout Their Lifecycle

Keywords: Application Ontology, Task-based Ontology Evaluation, Ontology Revision, Semantic Annotation, Ontology Lifecycle, Crowdsourcing

This paper presents the basis of our approach for evaluation of application ontologies. Adapting an existing task-based evaluation, this approach explains how crowdsourcing, involving application users, can efficiently help in the improvement of an application ontology all along the ontology lifecycle. A real case experiment on an application ontology designed for the semantic annotation of geobusiness user data illustrates the proposal.

INTRODUCTION

Ontology development is becoming a common task, which is nowadays not just a matter for ontologists. In the literature many ontology development methodologies have been proposed to help nonexperts build their own ontologies such as [START_REF] Noy | Ontology development 101: A guide to creating your first ontology[END_REF], [START_REF] Sure | OntoEdit: Collaborative ontology development for the semantic web[END_REF], [START_REF] Sure | Ontology engineering methodology[END_REF] and [START_REF] Suarez-Figueroa | The NeOn methodology for ontology engineering[END_REF]. However, according to [START_REF] Neuhaus | Towards ontology evaluation across the life cycle[END_REF], currently, there is no agreement on a methodology for development of ontologies, and there is no consensus on how ontologies should be evaluated.

As said by [START_REF] Brank | A survey of ontology evaluation techniques[END_REF], ontology evaluation is the problem of assessing a given ontology from the point of view of a particular criterion of application. It could help ontology developers evaluating their results and possibly guiding the construction process and any refinement steps. According to [START_REF] Vrandečić | Ontology evaluation[END_REF] this would make them feel more confident about their results, and thus encourage them to share their results with the community and reuse the work of others for their own purposes. Though, because of the lack of a consensus, evaluation techniques and tools are not widely utilized in the development of ontologies [START_REF] Neuhaus | Towards ontology evaluation across the life cycle[END_REF]. This can lead to ontologies of poor quality and is an obstacle to the successful deployment of ontologies as a technology.

In this paper we focus on the evaluation of application ontologies. Application ontologies describe the domain of specific applications [START_REF] Malone | Reference and Application Ontologies[END_REF]. They are built from scratch to make it stick to applications specific requirements. Consequently a pertinent evaluation consists in assessing their effectiveness against the different tasks they have to solve within the application for which they have been built [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF]. This evaluation step is crucial to guide their refinement. But in practice it is often skipped. It may be due to the difficulty of distinguishing which part of the application outputs really depends on the ontology itself and not on the application. Also few studies have addressed the evaluation of application ontologies through their specific uses within the application [START_REF] Brewster | Data driven ontology evaluation[END_REF]. Therefore we attempt to promote the systematic effectiveness evaluation of application ontologies by proposing a simple adaptation of the task-based approach of [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF] using crowdsourcing all along their lifecycle.

The rest of the paper is articulated in 4 sections. The second section presents a state of the art on application ontologies and ontology evaluation and the related works. The third section presents our proposal and an application experience. The fourth section presents our conclusions. This section introduces the related works about application ontologies, ontology evaluation and the deadlocks identified for the evaluation of application ontologies.

Application Ontologies

According to [START_REF] Malone | Reference and Application Ontologies[END_REF], an application ontology is an ontology engineered for a specific use or application focus and whose scope is specified through testable use cases. Application ontologies usually reuse, derive or reference recognized ontologies to construct ontological classes and relationships between classes [START_REF] Shaw | Generating application ontologies from reference ontologies[END_REF]. According to [START_REF] Guarino | Formal ontology in information systems: Proceedings of the first international conference[END_REF], this topdown approach promotes the reuse of ontologies. However, in practice, building reusable ontologies is a costly process. Consequently a frequent alternative consists in building application ontologies from scratch and then generalizing them to domain and task ontologies (bottom-up approach). For instance a bottom-up approach called Goal-Oriented Application Ontology Development Technique, presented in [START_REF] Santos | A Case Study on the Construction of Application Ontologies[END_REF], has been designed to guide the development of application ontologies from the explicit specification of their application goals translated into rules and facts. Generally, both top-down and bottom-up approaches require well defining the tasks the application ontology has to solve through the application. These tasks can be multiple (conceptual similarity calculation, disambiguation, knowledge extraction, semantic annotation, etc.) but are generally closely related to the application processes.

Ontology Evaluation

According to [START_REF] Gómez Pérez | Evaluation of ontologies[END_REF], ontologies, such as any other resources used in software applications, should be evaluated before (re)using it in other ontologies or applications. Evaluation of the ontology content, i.e. its concepts definitions, its taxonomy and its axioms, as well as evaluation of the software environment are therefore critical before integrating them in final applications. Many ontology evaluation methodologies have been proposed since 1995, among those, the well-known works of (Gómez-Pérez, 1995) and [START_REF] Guarino | Evaluating ontological decisions with OntoClean[END_REF]. In 2005, [START_REF] Brank | A survey of ontology evaluation techniques[END_REF]'s survey identified the main ontology evaluation approaches types: those based on comparing the ontology to a gold standard [START_REF] Maedche | Measuring similarity between ontologies[END_REF], those based on using the ontology in an application and evaluating the results [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF], those involving comparisons with a source of data about the domain to be covered by the ontology [START_REF] Brewster | Data driven ontology evaluation[END_REF], and those where evaluation is done by humans [START_REF] Lozano-Tello | Ontometric: A method to choose the appropriate ontology[END_REF]. In addition, authors grouped these approaches based on the level of evaluation: vocabulary level, taxonomy level, semantic relations level, context or application level, syntactic level, and structure, architecture and design level evaluation. They are all suited for the three first levels but only human-based evaluation can cover the three other levels. Several issues are still addressed today, such as the need to have a detailed methodology to allow performing evaluation throughout the entire ontology lifecycle (Staab & Studer, 2013). Also evaluation of application ontologies has been little addressed [START_REF] Malone | Reference and Application Ontologies[END_REF].

Application Ontology Evaluation

Authors of [START_REF] Malone | Reference and Application Ontologies[END_REF] state that application ontologies should be evaluated against a set of use cases and competency questions, which represent the scope and requirements of the particular application. This approach, called application-based evaluation, is founded on the fact that the outputs of the application or its performance on a given task might be better or worse depending partly on the ontology used in it. According to [START_REF] Brank | A survey of ontology evaluation techniques[END_REF], one might argue that a good ontology is one, which helps the application in question produce good results on the given task. However authors identify some drawbacks concerning application-based evaluation approaches:

(1) as an ontology is good or bad considering a particular task, it is difficult to generalize the approach, (2) if the ontology is only a small component of the application, its effect on the output may be relatively small and indirect, (3) comparison of different ontologies cannot be handled if they cannot all be plugged in the same application. In [START_REF] Vrandečić | Ontology evaluation[END_REF] the author minimizes these drawbacks by stating that ontologies are often tightly interwoven with an application, and, that the user never accesses an ontology directly but always through this application. Therefore the application often needs to be evaluated with the ontology, regarding the ontology as merely another component of the used tool. He adds that such a situation has the advantage that well-known software evaluation methods can be applied.

The Task-based Approach

One of the major works addressing applicationbased ontology evaluation is the task-based approach of [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF]. This approach provides an evaluation scheme on three basic ontological levels: vocabulary (concepts), taxonomy and semantic relations. For these three levels, the authors define three shortcomings the evaluation results have to show: insertion, deletion and substitution errors. Insertion errors indicate superfluous concepts, isa-and semantic relations, deletion errors indicate missing ones and substitution errors indicate off-target ones. Then given appropriate tasks and maximally independent algorithms operating on the ontology in solving these tasks, and, given a gold standard, this approach allows calculating the error rates corresponding to specific ontological shortcomings at each ontology level. A gold standard is the set of "perfect" outputs the task is expected to provide on the corpus on which it is run. Therefore a gold standard must be defined for each task as such as the explicit definition of the tasks, the ontology, and the application. A task is required to be sufficiently complex to constitute a suitable benchmark for examining a given ontology. The performance results must substantially depend on the way relations are modelled within the ontology. One ontology is sufficient as it is evaluated in terms of its performance on a given task within the application. The application is defined as a specific algorithm that uses the ontology to perform the task. However the algorithm can have more or less influence in the application output and it is sometimes difficult to distinguish which of the algorithm or the ontology is in cause. Therefore to obtain meaningful performances results, the algorithm must be kept constant within the ontology evaluation/revision cycle. Once all these components are defined, the evaluation can be launched. Then the revision phase of the ontology simply consists in undertaking the changes corresponding to the to the identified errors before running another evaluation round.

In the next section, we propose an adaptation of this approach taking into account user contributions to improve the evaluation accuracy all along the ontology use.

PROPOSAL

This section presents the basis of our proposal and illustrates it through a real case experiment.

A Task-based Evaluation through Ontology Lifecycle

Our approach consists in adapting the task-based approach of [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF]) by delegating the evaluation job to the users, using the application as a crowdsourcing platform, instead of using a gold standard. In [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF], the gold standard is actually produced by means of annotators (trained by humans) agreeing on mutually satisfactory solutions for the cases of disagreement.

Therefore its accuracy clearly depends on the performance of these annotators. Besides, if the application domain changes and the ontology needs to evolve, a new gold standard has to be produced, which is not convenient to evaluate an ontology during its entire lifecycle. Instead, a crowdsourced evaluation, in which users participate, can be realized all along the application lifecycle.

Crowdsourcing is defined in [START_REF] Hosseini | The four pillars of crowdsourcing: A reference model[END_REF] as a business model, where tasks are accomplished by a general public, called the crowd. This model has recently been promoted for the domain of information systems analysis and design namely through the involvement of users in evaluating the software [START_REF] Ali | Social adaptation: when software gives users a voice[END_REF] [START_REF] Pagano | User involvement in software evolution practice: a case study[END_REF].

In our approach, we consider the application users as the crowd, the evaluation procedures as the crowdsourcing tasks and the application as the crowdsourcing platform.

Our task-based evaluation methodology comprises the 6 following steps: 1. Definition of the application used as a crowdsourcing platform: distinction between tasks driven by the ontology and other tasks, 2. Definition of the (consistent) ontology: domain and scope, role within the application, the level of contribution regarding each task it drives, 3. Definition of the tasks driven by the ontology. 4. For each task, choice of the ontology levels to evaluate : vocabulary, taxonomy, semantic relations.

5.

For each ontological level, definition of the corresponding error types: deletion, addition, and substitution errors.

6. For each task to evaluate, definition of the crowdsourcing task: explicit process each user participating in the evaluation has to follow, guidelines for the qualification of the tasks outputs and their evaluation w.r.t. error types, indications for potential revision. Within this step users can contribute to the ontology revision by proposing a change (ex: addition of a missing concept, etc.). Therefore if a majority of users agrees on a proposed revision, it can be translated into ontological change and applied on the ontology. Then a reasoner can assess the ontology consistency after the application of the change in order to decide to commit or rollback it. Inversely if a consensus cannot be reached, the different points of view can be taken into account by applying each suggestion of change one by one until the ontology is no more consistent.

We believe this approach has several advantages. First, the ontology effectiveness is tested in the real production environment by real users on real data. Second, evaluation can be done during the application use if the users are given the ability of revising the tasks outputs; then a decision algorithm can be implemented to assess the user suggestions and translate them into ontological changes [START_REF] Klein | Change management for distributed ontologies[END_REF]. Third as users point of view on the application domain can change over time, they may revise application outputs they used to consider correct. In this case the ontology is able to evolve and stay up-to-date with the application community of users all along its lifecycle.

Experience of Semantic Annotation Task Evaluation

Here we describe the application of this methodology on a real case experiment.

1. The application considered is the SaaS geobusiness decision software, CD7Online, within which users build maps by processing and representing statistical data on geographical basemaps stored in their workspaces.

2. The ontology considered describes the CD7Online application domain including descriptive statistics data files (tables of data), geographical data files (base maps) and maps projects (organization charts), but also the platform related business processes and uses. It is specified with the OWL DL language and contains about 10000 axioms.

3.

The ontology has been developed to drive a task of semantic annotation, supporting a recommender system suggesting users relevant data and processes.

This task allows extracting metadata such as geographical level, year, theme, statistical indicator, etc., from the user data workspaces and to represent their relations with user data within RDF annotations. Built on the top of a triplestore containing these annotations a visualization tool, available in the form of a graphical 2D interactive graph (cf. Fig. 1) allows users intuitively browsing their own workspaces, navigating through the annotations, starting from their "Home" (cf. "Mes cartes et mes données" on Fig. 1 step 1). We chose to use this existing tool to establish and perform the evaluation procedure.

4. The evaluation Level: Like in [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF]) performance of the ontology can be evaluated at the semantic relation level as annotations are semantic relations between data and metadata instantiated from the ontology model.

5.

Three semantic relation error types are defined. A correct annotation corresponds to a correct nontaxonomic relation between a data and a metadata.

An annotation with one of the following errors is assessed incorrect: missing annotation (deletion), superfluous annotation (insertion), and wrong annotation (substitution).

6. The evaluation procedure given to the users is articulated in 5 steps: (1) choosing a data to search, (2) navigating in the graph according to the metadata they consider the most related to this data, (3) assessing the accuracy, (4) if not accurate, identifying the error type and eventually (5) proposing a revision. During the process, if the users manage to find it within the first direct path they take within the graph, the corresponding annotation of this data is considered accurate. If they need to use at least another path to find it, it means the annotation corresponding to the expected data is missing, this is a deletion error. If they find the data but if this data is only related to a wrong metadata, this is a substitution error. And if they find an unexpected metadata linked to the data in addition to accurate ones, it is an insertion error.

Fig. 1 describes an application example of the process. The first step shows the default graph on which the user can see the metadata extracted from his workspace (cf. Fig. 1 step 1). If the user considers his data is related to population, he selects the "th_population" category. Second, the graph morphs to show the statistical indicators related to this category (cf. Fig. 1 step 2). Among these, the user chooses the "Femme" indicator (i.e. "Woman"), because the data he is looking for is related to a population of women. Third, the graph morphs to show the statistical data corresponding to this indicator (cf. Fig. 1 step 3). The user finds the data "Femmes chômage" (i.e. "Unemployed women"), and validates the annotation.

Results

A first evaluation round, conducted on a sample of ten voluntary users, showed a percentage of 35,2% accurate annotations with statistical indicators obtained on a set of 571 real data series (against 38,4% deletions, 26,5% substitutions, 0% insertions). These results showed the ontology needed to be enriched and refined to obtain better outcomes. After the revision guided by these results, the percentage of accurate annotations resulted from a second evaluation conducted on a sample of ten other users, reached 82,8% of good annotations.

CONCLUSIONS

This paper presented a simple task-based ontology evaluation approach adapted from [START_REF] Porzel | A task-based approach for ontology evaluation[END_REF])'s methodology. Dedicated to application ontologies, it aims at facilitating evaluation and revision of application ontologies during their entire lifecycle, by delegating these tasks to voluntary users given an explicit procedure. Following (Staab & Studer, 2013)'s recommendation, evaluation can be done all along the ontology lifecycle and fit to the users' evolving points of view. Also first results of the experiment on a semantic annotation task, conducted on small samples of voluntary users, are promising. Today we are working on extending this evaluation experience to a true crowdsourcing one, by opening it to all users of CD7 in order to assess the relevance of the approach across the application lifecycle. The next step will consist in automating the inclusion of users revision suggestions by implementing a decision algorithm and translating the results to ontological changes. A future step would be to generalize this approach on different types of tasks in order to establish template procedures.

Figure 1 :

 1 Figure 1: Data search test within the interactive semantic visualization of a working group.


teaching and research institutions in France or abroad, or from public or private research centers.

Athena and Pallas, Image, Copies, Fakes and Doubles Françoise Létoublon

The question of the fake in visual arts may be still more complicated than in literature. It raises three aspects of the fake, in reference to the original author, to the time of creation, and to the resemblance between original and copy. Greek literature offers a great amount of allusions and anecdotes on a statue called Palladion Goddess Athena's surname Pallas, that played an important part in the legend of Troy and in several foundation myths of cities around the Greek and Roman ancient world. We shall try to analyze some aspects of the question of the fake or genuine Palladion in myth. Athena's statue named Palladion was not mentioned in Homer, but the important part it played in the legend of Troy appears in numerous other texts, which seems to imply that it was an key of first importance to victory or defeat in the war in several parts of the Epic Cycle. Thus the hypothesis that it was alluded to in the Iliad being explicitly mentioned appears sustainable. The paper aims to detect the ancient allusions or references to a very complicated story and to look at the iconographical testimonies as well1 . The task is not easy, since texts, though numerous, are generally recent, but some images undoubtly ancient prove that the legend circulated early. The complications of this story might imply that the statue had some secret powers, and therefore required to stay hidden in the adyton, maybe even that uttering its name was forbidden: the scarcity of ancient literary documents could then find an explanation, but of course, it cannot be demonstrated.

The most detailed narratives occurs in Apollodorus, Bibl.3.142-146 and in Dionysius Halicarnassensis. Let us begin with Ps. Apollodorus. The passage is clearly articulated in two parts, the first one concerning the presence of the statue in Troy, the second one telling its divine origin. Though this succession may appear surprising, we shall keep in trust to it, since it will allow us to begin the research with the statue moving in the human world and to close it with its creation by a goddess as a copy and substitute of a beloved person, which might imply that fabricating a statue is generally meant as the creation of a fake imitation of a figure, be it human or divine. In between, with the help of other authors, historians or poets, we shall see other aspects of the statue.

Paragraphs 3.142-3 of the Bibliotheca recall Ilos' travel from Greece to Phrygia. He was chosen as king and received an oracle telling him to follow a cow and to found a city in the place where the cow would stop, which shows a strong parallel to the better known Theban myth. He followed the instructions and founded a city named Ilios his own name. He asked Zeus to give him a sign, and a statue fell from the skies before Ilos' tent: the Greek text says τὸ διιπετὲς παλλάδιον πρὸ τῆς σκηνῆς κείµενον ἐθεάσατο : "he admired the διιπετὲς palladion ». Which meaning does the adjective διιπετὲς convey? Is it semantically appropriate? We will see further that it is probably a formulaic epithet,2 which may thus appear in more or less pertinent contexts. Let us start from formal facts: it is a compound adjective, the first element of which Διο-(a variant form Διιπετὲς will be met later) can only be a genitive form of origin (or in the case of Διι-a dative with an instrumental value) of the name of Zeus. The meaning is then: "fallen from Zeus" or "by Zeus" (instrumental value of the dative). The epithet was no longer understood, probably already at the time where the Bibliotheca was composed, and it was translated in Latin as if it meant "fallen from the sky". Anyhow, the epithet seems here quite appropriated, since the object was sent by Zeus in response to Ilos' prayer.

In a more detailed narrative, Dionysius of Halicarnassus tells in great details the foundation of Rome through ancient Pelasgians, Aborigenians and Trojans who escaped Ilios with Eneas during the city's fall. In 1.67, he ventures to explain how the Roman images of the gods were taken from Troy to Italy (first to Lavinium, later to Alba) by Eneas and kept there in a temple, and once deplaced to a secret place (ναοῦ χωρίον ἔχοντος ἄβατον); in the following night, the statues moved by themselves and were found standing on their former bases (διαµείψαντα τὰ βρέτη τὴν στάσιν ἐπὶ τῶν ἀρχαίων εὑρεθῆναι κείµενα βάθρων); the same phenomenon occurred once again. Then it was decided to leave the statues in their place and to affect six hundred peoples caring for them. The historian gives their names, and particularly the Roman one (Penates). He ends paragraph 67 with a remark on the absolute necessity of secrecy around them, and in general:3 it is forbidden to see them and even to speak or write about them; we have to remember this obligation, which may have been available for the sacred objects from the first time. In the following paragraph, he mentions a small temple on the forum in Rome, in a place called in Velia, where Trojan gods' images could be looked at by everybody, two sitting youths with spears and military equipment. Among the sources about them, he explicitly mentions Arctinos who is known otherelse as one of the authors of the Epic Cycle, namely of the Aithiopis and the Iliou Persis. 4 The palladion occurs in this context of the sacred objects brought with them by Greek migrants first to the island of Samothrace, then to several places in Phrygia, then to Italy, and eventually to Rome: Dionysius actually uses the plural palladia, which raises the issue of a genuine Palladion versus a fake Palladion. The detailed narrative of 1.68.3 reads thus: Chryse, daughter of Pallas (note that for Dionysius, this Pallas is a man) married Dardanos and received as marriage gifts from Athena the Palladia and other sacred objects of the Great Gods, since she had been initiated to the Mysteries: 5 Χρύσην τὴν Πάλλαντος θυγατέρα γημαμένην Δαρδάνῳ φερνὰς ἐπενέγκασθαι δωρεὰς 'Αθηνᾶς τά τε Παλλάδια καὶ τὰ ἱερὰ τῶν μεγάλων θεῶν διδαχθεῖσαν αὐτῶν τὰς τελετὰς. Afterwards, escaping the flood (τὴν ἐποµβρίαν φεύγοντες), Dardanos and his Arcadians flew from the Peloponnese to Samothrace, where they founded the Mysteries. 6 Dardanos seems then to be the main chief in the initiation process: ἐν δὲ τῇ Θρᾳκίᾳ νήσῳ τοὺς βίους ἱδρύσαντο, κατασκευάσαι τὸν Δάρδανον ἐνταῦθα τῶν θεῶν τούτων ἱερὸν ἀρρήτους τοῖς ἄλλοις ποιοῦντα τὰς ἰδίους αὐτῶν ὀνομασίας καὶ τὰς τελετὰς αὐτοῖς τὰς καὶ εἰς τόδε χρόνου γινομένας ὑπὸ Σαμοθρᾴκων ἐπιτελεῖν.

He did not stay in the island, but left for Asia Minor with the Palladion and other sacred objects. 7 Thereafter, Dionysius quotes the text of the oracle that mentions the aporthetos polis (then Dardania, later Ilios nearer to the coast): the statue was meant to guarantee the safety of the city forever (note the formulaic expressive reduplication in the text of τὸν ἀεὶ χρόνον and ἤµατα πάντα). 8 The well attested links between oracles and formulaic diction in Greek 5 Those Mysteries of the Great Gods are thus supposed to come from Peloponnese.

6 The secret of the Mysteries of Samothrace have been so well kept that we do not know much about them, see 0000.

7 Dionysius' text may contain contradictions, since it seems to imply that the same sacred objects followed Dardanos in his migrations from Peloponnese to Phrygia through Samothrace, whereas the Mysteries of Samothrace lasted long after Dardanos and others left for Asia. Dionysius actually makes a distinction between the sacred objects proper to Samothrace and other ones: ὡς δὲ μετῆγε τοῦ λεὼ τὴν πλείω μοῖραν εἰς τὴν 'Ασίαν τὰ μὲν ἱερὰ τῶν θεῶν καὶ τὰς τελετὰς τοῖς ὑπομείνασιν ἐν τηῆηῇ νήσῳ καταλιπεῖν•• τὰ δὲ Παλλάδια καὶ τὰς <τῶν> θεῶν εἰκόνας κατασκευασάμενον ἀγαγέσθαι μετ'ʹ αὐτοῦ.

8 The formula ἤµατα πάντα is obviously Homeric: Il. 8. 539, 12.133, 13.826, 14.235, 276, 19.226, 23.594, 24.491, Od. 2.55, 205, 4.209, 592, 5.136, 210, 219, 6.46, 7.94, 257, 8.431, 468, 9.123, 10.467, 15.54, 17.534, 21.156, 23.6, 336, 24.24, it occurs very often in the verse end, but not constantly. The phrase τὸν ἀεὶ χρόνον appears as a gloss or comment, as if the late author was aware that Homeric Greek could confuse. For more clarity, we quote here the whole text of the oracle as it stands in Roman Antiquities 1.68.4:

Εἰς πόλιν ἣν κτίζῃσθα θεοῖς σέβας ἄϕθιτον αἰεὶ θεῖναι, καὶ ϕυλακαῖς τε σέβειν θυσίαις τε χοροῖς τε.

tradition met here allow to conclude that oral tradition helped preserving the words of a very old text. 9 We are then in Troy or Troad in the times of its occupation and foundation. Let us remark that those migrants are supposed to be Greek: therefore, the war of Troy does not consist at all for Dionysius, who is himself of Ionian origin, in a clash of Greeks against Asiatic enemies: it is rather the issue of Greek tribes fighting one another because of women raped with booty and wealth. ἐκ τῆς πόλεως καὶ ἐλθεῖν ἄγοντα εἰς 'Ιταλίαν.

After having thus mentioned in prose the importance of the objects sent by the gods (θεόπεμπτά) for keeping the city safe (καὶ σωτηρίας κύρια τῇ πόλει),12 he very quickly resumes the taking of the downtown part of Troy with an absolute genitive (ἁλισκομένης δὲ τῆς κάτω πόλεως), which explains that Eneas could take the sacred objects from Acropolis and flee with them to Italy.

Thus appears the existence of two Palladia: Dionysios invokes the authority of Arctinos (is it a quotation or rather a translation in prose?) for an explanation: Zeus gave Dardanos one Palladion, and it is this object that was kept in Ilios until its fall: 'Αρκτῖνος δέ ϕησιν ὑπὸ Διὸς δοθῆναι Δαρδάνῳ Παλλάδιον ἓν καὶ εἶναι τοῦτο ἐν 'Ιλίῳ τέως ἡ πόλις ἡλίσκετο κεκρυμμένον ἐν ἀβάτῳ•• But a copy of the original Palladion had been made, so close to the genuine image that people who saw it could be abused. This image was exposed to gaze, and the (Achaean) thieves took the copy, the fake Palladion instead of the genuine one: εἰκόνα δ'ʹ ἐκείνου κατεσκευασμένην ὡς μηδὲν τῆς ἀρχετύπου διαϕέρειν ἀπάτης τῶν ἐπιβουλευόντων ἕνεκεν ἐν ϕανερῷ τεθῆναι καὶ αὐτὴν 'Αχαιοὺς ἐπιβουλεύσαντας λαβεῖν. The end of paragraph 69 precises that the genuine Palladion taken from Troy by Eneas was still kept in Dionysios time in 'Hestia's' temple, which refers to Latin Vesta, with the city Fire, and he announces that he will come back to this issue later (i. e. 2.66---7: see below).

The same author comes back later to the Palladion in the wake of the Roman heritage and tradition: the temple of Hestia preserves not only the well known sacred public fire, but also the objects brought by Dardanos from Samothace, later by Eneas from Troy to Italy: τοῦτο δὴ λαβόντες ὁμολογούμενον ἐπισυνάπτουσιν αὐτοὶ στοχασμούς τινας ἰδίους, οἱ μὲν ἐκ τῶν ἐν Σαμοθρᾴκῃ λέγοντες ἱερῶν μοῖραν εἶναί τινα ϕυλαττομένην τὴν ἐνθάδε, Δαρδάνου μὲν εἰς τὴν ὑϕ'ʹ ἑαυτοῦ κτισθεῖσαν πόλιν ἐκ τῆς νήσου τὰ ἱερὰ μετενεγκαμένου, Αἰνείου δὲ, ὅτ'ʹ ἔϕυγεν ἐκ τῆς Τρωάδος ἅμα τοῖς ἄλλοις καὶ ταῦτα κομίσαντος εἰς 'Ιταλίαν, οἱ δὲ τὸ διοπετὲς Παλλάδιον ἀποϕαίνοντες εἶναι τὸ παρ'ʹ 'Ιλιεῦσι γενόμενον, ὡς Αἰνείου κομίσαντος αὐτὸ δι'ʹ ἐμπειρίαν, 'Αχαιῶν δὲ τὸ μίμημα αὐτοῦ λαβόντων κλοπῇ•• (2.66.5) Thus there are for him two divergent traditions, one admitting that Dardanos took the whole of the sacred objects from Samothrace, whereas the other one the Palladion, qualified as διοπετὲς, was sent by Zeus: the coincidence with Apollodorus' τὸ διιπετὲς Παλλάδιον is striking, maybe all the more with a very slight difference in the text, implying either a dative or a genitive of Zeus' name in the first term of the compound. 13 It seems then that for the first tradition, the Palladion was part of the marriage gifts given to Dardanos's wife by Zeus' daughter as seen above. In the same part of the narrative, Dionysios also alludes to the theft of the palladion, and to the fact that the Achaean thieves were abused and took a copy only (τὸ μίμημα αὐτοῦ).

The occurrence of the formula τὸ διοπετὲς παλλάδιον occurs in the context of the statue's presence in Rome. The narrative does not seem entirely consistent with Book 1 and the mention of Dardanos' wedding gift. But Dionysius is perhaps more coherent with the historical method here, where he recognizes that several mythological traditions may justify the state of things: in 1.69.4, he called the Palladion the legendary, τὸ µυθευόµενον παλλάδιον, and some anecdotes that he tells, in concordance with other authors, seem to imply that the statue had magical powers: in 2.66.4, Lucius Caecilius Metellus, seeing Vesta's temple burning, as the priestesses flew without the sacred objects, rushed into fire without caring for his own safety, and saved them. Dionysius does not mention any misfortune for this important Roman citizen, who received a reward for this courageous deed. But other saviors of the objects, and particularly the Palladion, were less fortunate, as some cases of blindness particularly testify. Plutarch, in his Minor Parallels, mostly describes Ilos' blindness with two variants. In one case, Ilos took (ἥρπασε may imply violence) 14 the statue out of the adyton to the daylight, and became blind (ἐτυφλώθη 15 ). 16 He probably had good intentions, since the temple was burning 13 Thus the variant in the tradition could be a strong clue to the authenticity of the formula coming from an oral tradition relative to the Epic cycle.

14

The verb ἁρπάζω bis currently used in Greek for the rape of women for instance.

15

The aorist may imply that it happened suddenly, as if the statue were punishing her raper. in both variants (καιόµενον). Plutarchus gives the reason of his blindness: it was not allowed to gaze at the statue (οὐ γὰρ ἐξῆν ὑπ' ἀνδρὸς βλέπεσθαι). According to Plutarchus, it was the version of Derkullos, whereas Aristeides of Miletos suggested that Ilos was trying to save the statue from burning. Anyhow he became blind all the same, but recovered his sight later. Plutarchus includes this tale in his Parallells to show that the Trojan Ilos had a fate similar to the Romans Metellus (mentioned in Dionysios Halicarnassensis) and Antylus. From those allusions, it can be deduced that the statue had the power, when taken to daylight, to throw dangerous lightning flashes that could blind human beings. The description of the statue in Virgil's Aeneid strikely corresponds to this supposition. 17 The Greek traitor Sinon tells for the Trojans that after they have stolen the Palladion, Diomedes and Ulysses encountered goddess Athena's wrath:

Tritonia gave signs of this by no uncertain portents. The statue had just been set down in camp. Dancing flames blazed from her upwards-turned eyes and salt sweat passed over her limbs. Three times she actually leaped from the ground (a wonder to relate), carrying her round shield and quivering spear. At once Calchas pronounced that the Greeks were to venture on the waves in flight; Pergama was not to be taken by Argive arms, if they did not seek out fresh auspices at Argos and bring back the goddess whom they had carried off with them by sea in their curved ships. 18 The fact that Sinon is lying in the context of Virgil's interpretation of the Trojan war does not impel the general knowledge of the brigthness of the statue, and its magical powers. In Quintus of Smyrna 10.350-60 is found an allusion to the Palladion in a crypted prediction: 19 the mention of Zeus (called here by the periphrase "son of Cronos") throwing the statue from the Olympos: the aorist with apocope and assimilation in the beginning of the verse is typically Homeric. 20 Quintus Smyraneus does not call the Palladion as such, but uses a periphrase which means "immortal apparence", and strengthens the Homeric tone of the passage. 21 So it seems that Quintus imitated Homer's style, perhaps in a particular instance when the content is not actually Homeric… It is also possible that the Palladion was not explicitely designated here, precisely because of the magical power of the statue: in Dionysius Halicarnassensis, the description of the foundation of the Samothrace Mysteries by Dardanos well implies the presence of some secret objects and words, 22 so that the periphrase used by Quintus might respect a language prohibition. To those classical sources, the New Jacoby online allows now to add the Papyrus Rylands 22a, which alludes to the theft of the Palladion by Odysseus and Diomedes, following a prediction given by the seer Helenus. Thus literary sources available on the Palladion are quite recent, but some of them include pieces of formulaic style that might refer to quotations of the Epic cycle or of oracular traditions transmitted from a period close to the Homeric epics. The formal variations appear strong testimonies to archaic remains kept in recent texts as linguistic fossils.

Graphic arts might also show some interesting testimonies. If we summarize Castiglioni's still unpublished study, the statue considered as the Palladion may have been linked to several episodes in the Epic cycle, namely Cassandra's rape by Ajax, the recovering of Helen by Menelaus in the Iliou persis his try to kill her, the theft of the Palladion mentioned by Proclus' summary as taking place in the Ilias parva, 23 and Aeneas carrying both 20

The TLG provides us with 8 occurrences of κάββαλ-: Il. 5. 343, 8.249, 9.206, 12.206, 23.127, 683, Od. 6.172, 17.302.

21

ἄµβροτον is used actually in Homer with αἷµα, θεόν, κρήδεµνον, ἱστόν; εἶδος appears in several Homeric formulas, none with an epithet. Thus both words are Homeric, their association is not.

22

References:

23

The first evidence is an Attic cup from the Hermitage Museum (LIMC 23), called the Makron cup (around 480 BC), with two Palladia. A Stockholm Amphora (LIMC 24), an Attic production of the same period (480), represents the same scene, but Athena replaces the Greek heroes between Diomedes and Odysseus who face each other and are both fully armed. Two Palladia also occur in an Apulian oenochoe from the Louvre (360-the Palladion and his father Anchises. The images showing two similar Palladia are very interesting, in comparison to the mentions in Dionysios of a copy of the Palladion, which was made on purpose as a piece of delusion against the enemies. We might perhaps call it an 'official fake'. Anyhow, if the legend admits the existence of two Palladia, a genuine one and a fake, whatever Diomedes and Odysseus took one and Eneas the other, it raises a serious issue: the oracles considered that the city would remain safe as long as the Palladion stay in its hiding place. If the Achaeans took the fake, the city should not have fallen, unless the oracles were lying. If they took the genuine, how could Eneas still have the genuine Palladion to bring to Italy? The argument of a second Palladion opens the way to sophistic speculations on fate and history, and maybe explains that from two, the number could be expanded to the infinite: it may have been the main reason why several cities in ancient Greece and Italy claimed that they owned a Palladion kept among their most precious heritage from the war at Troy. 24 Before going back to the fabrication of the statue, it seems necessary to insist on the status of the formulas and their relation to poetry and metrics, which could reveal quotations from the Epic cycle, particularly Διο/Διιπετὲς παλλάδιον. We know two Greek probable "quotations" (Apollodorus and Dionysius Halicarnassensis), and Latin "translation" in Dictys Cretensis (a caelo). The quotation is not explicit, and is not officially attributed to the Cycle either by one or by the other author. The fragments of the Cycle are very scarse, but they seem to use the same hexametric form as the epics, and generally Archaic poetry. Let us then try a metrical analysis: in the dactylic hexameter παλλάδιον begins with a long vowel followed by three short vowels: if the final o stands in front of a lengthening consonant, the word can be scanned -˘ ˘ / -Thus the nominative form of the word cannot be located at the end of the verse. In other cases of the flexion we can imagine παλλαδίῳor παλλαδίου:

-˘ ˘/ ˉˉ This form cannot stand at the end of the verse either, but it needs to be followed at least by another word supplying the brief vowels of the dactyl… On the other hand, διοπετὲς scanned -˘ ˘/ ˉˉ, implies anyway a metrical lengthening of the first iota. Thus the whole formula διοπετὲς (variant διιπετὲς in Apollodorus) παλλάδιον implies -˘ ˘/ ˉˉ and -˘ ˘/ ˉˉ , which cannot occur as a metrical sequence in the hexameter, but could reflect the dislocation of the terms with other words (something like διοπετὲς ˘ ˘ παλλάδιον ˘ ˘. The inversion of both terms does not provide a satisfactory solution. Parry's definition of the formula helps us understand Διοπετὲς (Διιπετὲς) as a fixed or particularising epithet (that is, which is only used to qualify this particular object in Homeric Greek): it gives the object the status of an individual, a person, providing it with almost a proper noun. We actually know that in Greek the onoma is first a proper noun, then only secondarily a "common" name. This settles the problem of the meaning of the epithet which becomes in Latin (and other existing translations of the Greek) "fallen from the sky ", while the first element of the compound Διοπετὲς (Διιπετὲς) can only be a genitive form of origin (or in the case of Διι-a dative with an instrumental value) of the name of Zeus. The meaning is then: "fallen from Zeus" or "by Zeus" (instrumental value of the dative).

The equivalence between Zeus and the sky is well known in the Indo-European reconstruction of the name (the correspondents skr dyauh, lat. dies"daytime" imply a close semantic reconstruction with the meaning 'bright day'. It is also valid to speak from a 'functional' point of view: the Iliad refers to the sharing of timai the sons of Cronos, ending with Zeus reigning over the sky and the earth, Poseidon over the sea and Hades over the underworld. Nethertheless it remains linguistically impossible in Greek to understand Διοπετὲς (Διιπετὲς) as meaning properly ἀπ᾽ οὐρανοῦor κατ᾽ οὐρανοῦ. On the contrary, the excerpt from Quintus quoted hereabove clearly indicates that Zeus threw the Palladion (Κρόνιων / κάββαλεν) from the top of the Olympus (ἀπ᾽Οὐλύµποιο). We assume that both details were mentioned in the earliest text, and that the actual meaning of διιπετές, as proved by the occurrence of such syntagms as Δία ῥῖψαι (Bibl. .145) and Κρόνιων /κάββαλεν (Quintus of Smyrnae 10.359-60), must be "thrown by Zeus". But as some late lexicographers attest, this phrase could no longer be understood during their times, and required a gloss saying "fallen from the sky" or mentions such as ἀπ᾽Οὐλύµποιο to replace it. P. Rylands 22 τὸτῆς ᾽Αθηνᾶς οὐράνι[ον ἄγαλµα, « Athena's celestial statue » seems consistent with this view.

It is Zeus who threw the Palladion from the Olympus, and Dardanos or Ilos who found it at his door (we will not dwell upon whether it was a wedding gift for Dardanos). 25 Zeus actually never uses epiphanies to show himself to mortals on earth as other gods do (Athena, Hermes, Apollon, Poseidon, Ares in the Epics). 26 In any case, in Homer's texts, Zeus always sits in the heights, on Mount Olympus or Mount Ida, and only leaves them for the sacred Tables of the Ethiopians; he sometimes stands up to brandish the lightning or the thunder, and to throw objects from his throne such as the rain of blood he sends out in the episode of Sarpedon's death. 27 Cf. The formula αἷµα διοπετές 'blood fallen from Zeus' met in the Homeric corpus (Il. 5.339, 5.870).

Apollodorus 3.144-5 refers to the statue of Pallas made by Athena after the death of her friend (who was Triton's daughter). 28 This statue was endowed with the attributes of the goddess (δόρυ, ἠλακάτη, ἄτρακτος, and the aigis which caused Pallas' death). Δία ῥῖψαι appears a development of or verbal equivalent for διιπετὲς.

4) the Palladion as a talisman

We will try now to show that the Palladion is a magical object, which was to our opinion largely underestimated in Faraone's book on the question. Given by the divinity, it must be kept inside (in the adyton of the temple, inside the city), far from the light. In daylight, the statue emits flashes of light ( Virgil, En. 2.) and can blind the person who sees it (cf. the blinding of Ilos as told by Derkyllos in Plutarch's Minor Parallels, and the Roman parallels he mentions in Roman tradition: Metellus etc.). This could be the reason why Diomedes and Ulysses steal the Palladion at night. The formula we analyzed probably comes from a fragment of the Little Iliad or the Iliupersis, as the summary by Proclus, and perhaps some iconographical representations, seem to indicate. But we must be careful : the theft of the Palladion could belong to the episodes of the fall of Troy, but the fall of the Palladion on Ilos or Dardanos's front door might belong to a previous episode. Naturally, if Διοπετὲς Παλλάδιον is a formula, it might be repeated in several episodes, without a deep semantic value or relevance in every one of them. In any case, Apollodoros and Dionysios of H. probably still had knowledge of the traditions resulting from the Cycle, in which the small statue of Athena (see the size accurately indicated by Apollodorus) had a formulaic epithet, but its meaning was forgotten over time.

Several texts and images show two Palladia and use the plural. The most explicit version is found in Dionysus Halicarnassensis who recalls Dardanos' travel from Arcadia to Samothrace. He seems to have received the statue there, either as a

27

The meteorological verbs are impersonal in Greek, but the imperative used in prayers asking for rain and addressed to Zeus, prove that rain and snow were thought as the result of his will and action.

wedding gift or rather "fallen from Zeus". Did he then have a copy of the statue made before leaving the island for Asia Minor with sacred objects, including the Palladia ? Later on, several episodes provide references, which are more or less explicit, to at least two Palladia: it may be inferred that the true or original statue came from Zeus who threw it on earth. The copy might have been designed as a trap for thieves who wanted to steal the statue for its magical power. Several of our texts refer to predictions linked to the Palladion (see Dionysius Halicarnassensis and Quintus Smyrnaeus mostly). The theft of the statue by Odysseus and Diomedes is a consequence of such a prediction: they knew that Troy would not fall if the Palladion remained in its adyton, far from human eyes. The allusions to their expedition are numerous, but Dictys Cretensis is the only one to provide an explanation for the method used by the Achean heroes: Antenor and his wife Theano secretely agreed to steal the statue from the temple and exchange it for a great amount of money. 29 We find this version very convincing because it could account for several details found in the Iliad and still unexplained: in Book 3, Antenor describes Odysseus' sublime eloquence in the famous Teichoscopia, comparing his words to a snow storm. 30 Was Antenor referring to his own experience of being swept away by this verbal storm? In Book 6, Hector asks his mother to offer a sacrifice to Athena. She brings her most splendid veil as a gift to be offered to the goddess by her priestess Theano. The sacrifice seems in perfect accordance to the ritual, but the goddess denies it (Il. 6.311 ... ἀνένευε δὲ Παλλὰς Ἀθήνη): this can be considered as a very bad omen for the city of Troy. One could assume that, even if she is a woman, Hecuba had no right to enter the adyton: therefore she asked the priestess to take care of the prayer and the offering on her behalf. But on the other hand, Athena cannot accept a veil coming from Paris's treasure chest. Is the statue mentioned in this occurrence the palladion or another cult object? The question remains open. Later on, some authors thought that Diomedes and Odysseus stole two statues (the palladia from Dardanos' time or later on); others suggested that they stole only one statue, but that is was the fake. It then could be inferred that Antenor and Theano were more clever than the polymetis Odysseus himself, since they sold a fake statue for a high price. 31 Anyway, the legend of the Trojan origins of Rome is based, among other elements, on the idea that Aeneas carried away with him, apart from his father on his back, his wife, his son, the Penates, and the original Palladion thrown by Zeus: this statue was then kept by the Romans in Vesta's temple, if we follow Dionysius Halicarnassensis.

The story takes on a sophistical turn that, as far as we know, has never been commented by ancient authors: if the statue was a guarantee of safety for the city, and if Diomedes and Odysseus took only a fake Palladion, a copy, and the original remained in the temple, why did Troy fall after all? The Palladion is key to the Achaeans's victory, and they will replace it with the Wooden Horse, another trap reversing the move from outside to inside. Let us come back to the text of the Bibliotheca and its narrative of the statue's creation by Athena. According to the mythograph, the goddess as a child, was raised by Triton, 32 with his daughter Pallas. Both of the girls used to play war games (ἀσκούσας τὰ κατὰ πόλεµον) and once came to quarrell. As Pallas was going to strike Athena, she was frightened. Zeus also was afraid for her and he streched the aigis before her as a protection. Pallas, startled, looked up, was striken by Athena and fell. Athena was deeply sorry (περίλυπον ἐπ'ʹ αὐτῇ γενοµένην), she made an image close to her (ξόανον ἐκείνηc ὅµοιον καταcκευάcαι), she put the aigis she was afraid of around her shoulders and placed it besides Zeus to be honoured there (καὶ περιθεῖναι τοῖc cτέρνοιc ἣν ἔδειcεν αἰγίδα, καὶ τιµᾶν ἱδρυcαµένην παρὰ τῷ Διί). But later, Electra after her rape, sought refuge by the image, and Zeus threw it with Ate into the Ilian country (ὕcτερον δὲ Ἠλέκτραc κατὰ τὴν φθορὰν τούτῳ προcφυγούcηc, Δία ῥῖψαι µετ᾿ αὐτῆc/ ᾽Ατης 33 καὶ τὸ παλλάδιον εἰc τὴν Ἰλιάδα χώραν), Ilos built a temple for it and honoured it (Ἶλον δὲ τούτῳ ναὸν καταcκευάcαντα τιµᾶν). Almost every word counts in this short narrative: the statue is a xoanon, which means a wooden archaic statue, it is rather small in size (three cubits: tripekhus) that could easily fit in human arms. The statue is clearly conceived in the idea of replacing the dead friend and

The sophistic issue mentioned above seems still more acute in this case.Did Theano, as the priestess of Athena,betray her city and her goddess, or did she act skillfully, in collusion with her husband, deceiving the skillful Odysseus himself? In both cases, the city's fall shows their mistaken choice. 32 ϕασὶ γεννηθεῖσαν τὴν 'Αθηνᾶν παρὰ Τρίτωνι τρέϕεσθαι, The detail could explain the frequence of Tritogeneia as appellation of Athena in our texts, be they Greek or Latin.

allowing grieving and solace: this correspond to the current ideas met in Greek literature about the birth of the statues and paintings in general, as show the myths of Protesilaos and his wife Laodamia and the myth of Boutades' daughter: the dead's wife or the fiancée of a soldier tries to have at least an image of the beloved one. 34 In addition to this essential feature, we may note a childish tone that could show that making an imitation (ἐκείνηc ὅµοιον) of her friend may be also for young Athena a way of playing with a puppy. 35 Therefore the Palladion is first an image of the beloved Pallas and a solace for her death, and as an image, it is a substitute for the person. But still when playing with her living friend, Athena may be considered playing with her own double: she is a girl, more or less of the same age, and they play together war games with a strong aspect of competition. It is well-known that children often invent imaginaries companions to play with. In this story, young Athena had an actual friend who was a kind of double for her, and she replaces her when dead by an image, another kind of double. 36 We meet with the subtle notion of double in mythology studied by Maurizio Bettini in a paper published in 2004. He distinguishes two kinds of creation of doubles in Greek myths, that proceed either from defence or from attack. Here, Pallas is from the beginning a double for Athena, as her foster father Triton is a double and substitute for Zeus. Zeus and the aigis defend Athena against the potential violence of her double, and Athena makes an image of it, that could gilt as defence of virginity. 37 Fighting against her own double is a kind of challenge such as Athena likes to cope with. The question we wanted to pose is the analogy between the notions of fake, double,and image: after the death of 34 See mainly Bettini 1992 and Steiner 2001. 35 Bettini devoted a whole chapter to puppies in Antiquity. They are considered a symbol of virginity, which fits particularly well for the virgin goddess playing in her childhood with another girl close to her.

37

However the examples of Electra in this version, of Cassandra raped by Ajax although she sought refuge near the Palladion, show that its protection is not sufficient.

her double, Athena makes an image of her which has very strange powers, hence a strange story, full of mysteries and enigms, and this image provoked many fakes and delusions. It might be that Homer's silence on the topic could be part of the problem.

Annex

29

  Dionysios Halicarnassensis 1.46 could contain an allusion to this version : 'Ιλίου κρατηθέντος ὑπ'ʹ 'Αχαιῶν, εἴτε τοῦ δουρείου ἵππου τῇ ἀπάτῃ, ὡς 'Ομήρῳ πεποίηται, εἴτε τῇ προδοσίᾳ τῶν 'Αντηνοριδῶν εἴτε ἄλλως πως, [ …] 30 Il.3.222 ἔπεα νιφάδεσσιν ἐοικότα, see Létoublon 000 1994 ????.

  10 In 1.69, Dionysius Halicarnassensis explains how Dardanos set the objects in the eponym city founded in Troad, 11 and how they were later transferred to Ilios where he temple was built for them on the acro(polis), with a secret room (adyton) he explains that the palladion remaining was taken from the by Eneas after the fall of Troy, which leads him to mention that the other one was stolen by Diomedes and Odysseus:

	Δάρδανον μὲν ἐν τῇ κτισθείσῃ τε ὑϕ'ʹ ἑαυτοῦ καὶ ὀνομασίας
	ὁμοίας τυχούσῃ πόλει τὰ ἕδη καταλιπεῖν, 'Ιλίου δ'ʹ ἐν ὑστέρῳ
	χρόνῳ συνοικισθέντος ἐκεῖσε μετενεχθῆναι πρὸς τῶν ἐγγόνων
	αὐτοῦ τὰ ἱερά. ποιήσασθαι δὲ τοὺς 'Ιλιεῖς νεών τε καὶ ἄδυτον
	αὐτοῖς ἐπὶ τῆς ἄκρας καὶ ϕυλάττειν δι'ʹ ἐπιμελείας ὅσης
	ἐδύναντο πλείστης θεόπεμπτά τε ἡγουμένους εἶναι καὶ
	σωτηρίας κύρια τῇ πόλει. ἁλισκομένης δὲ τῆς κάτω πόλεως τὸν
	Αἰνείαν καρτερὸν τῆς ἄκρας γενόμενον, ἄραντα ἐκ τῶν ἀδύτων
	τά τε ἱερὰ τῶν μεγάλων θεῶν καὶ ὅπερ ἔτι περιῆν Παλλάδιον
	(θάτερον γὰρ 'Οδυσσέα καὶ Διομήδην νυκτός ϕασιν εἰς ῎Ιλιον
	ἀϕικομένους κλοπῇ λαβεῖν) οἴχεσθαί τε κομίσαντα [τὸν Αἰνείαν]
	ἔστ'ʹ ἂν γὰρ τάδε σεμνὰ καθ'ʹ ὑμετέρην χθόνα μίμνῃ
	δῶρα Διὸς κούρης ἀλόχῳ σέθεν, ἡ δὲ πόλις σοι
	ἔσται ἀπόρθητος τὸν ἀεὶ χρόνον ἤματα πάντα.
	9
	See the entries Formulaic Language and Oracular Language in the EAGLL (Létoublon, to appear). Among
	the formulas, let us also note σέβας ἄϕθιτον αἰεὶ and δῶρα Διὸς κούρης.
	10
	As Herodotus put it previously in fis first book.

  ἄµβροτον εἶδος, ἀπ᾽Οὐλύµποιο Κρόνιων κάββαλεν … We can notice here the parallel with Dionysius Halicarnassensis' reference of the oracle, and Virgil, Aeneid 2. A Commentary, Leiden, 2008: 11. See his commentary to these verses, ibid.: 165-73. The statue (called sacrem effigiem in v. 167) turned the eyes away from the thieves and sent flames (arsere coruscae / luminibus flammae arrectis, v. 172-3 is particularly expressive).

	16
	See Létoublon on Blindness, 2011 à vérifier.
	17
	Virg. Aen. 2.171-179
	18
	Translation N. Horsfall,

  Bibl. 145-6 ϕασὶ γεννηθεῖσαν τὴν 'Αθηνᾶν παρὰ Τρίτωνι τρέϕεσθαι, ᾧ θυγάτηρ ἦν Παλλάς• ἀµϕοτέρας δὲ ἀσκούσας τὰ κατὰ πόλεµον εἰς ϕιλονεικίαν ποτὲ προελθεῖν. µελλούσης δὲ πλήττειν τῆς Παλλάδος τὸν Δία ϕοβηθέντα τὴν αἰγίδα προτεῖναι, τὴν δὲ εὐλαβηθεῖσαν ἀναβλέψαι, καὶ οὕτως ὑπὸ τῆς 'Αθηνᾶς τρωθεῖσαν πεσεῖν. 'Αθηνᾶν δὲ περίλυπον ἐπ' αὐτῇ γενοµένην, ξόανον ἐκείνης ὅµοιον κατασκευάσαι, καὶ περιθεῖναι τοῖς στέρνοις ἣν ἔδεισεν αἰγίδα, καὶ τιµᾶν ἱδρυσαµένην παρὰ τῷ Διί. ὕστερον δὲ 'Ηλέκτρας κατὰ τὴν ϕθορὰν τούτῳ προσϕυγούσης, Δία ῥῖψαι µετ' αὐτῆς καὶ τὸ παλλάδιον εἰς τὴν 'Ιλιάδα χώραν, ῏Ιλον δὲ τούτῳ ναὸν κατασκευάσαντα τιµᾶν. καὶ περὶ µὲν τοῦ παλλαδίου ταῦτα λέγεται.] Bibliotheca, 142-146:῏Ιλοc δὲ εἰc Φρυγίαν ἀφικόµενος καὶ καταλαβὼν ὑπὸ τοῦ βαcιλέως αὐτόθι τεθειµένον ἀγῶνα νικᾷ πάλην• καὶ λαβὼν ἆθλον πεντήκοντα κόρουc καὶ κόραc τὰc ἴcαc, δόντοc αὐτῷ τοῦ βαcιλέωc κατὰ χρηcµὸν καὶ βοῦν ποικίλην, καὶ φράcαντοc ἐν ᾧπερ ἂν αὐτὴ κλιθῇ τόπῳ πόλιν κτίζειν, εἵπετο τῇ βοΐ. ἡ δὲ ἀφικοµένη ἐπὶ τὸν λεγόµενον τῆc Φρυγίαc Ἄτηc λόφον κλίνεται• ἔνθα πόλιν κτίcαc Ἶλοc ταύτην µὲν Ἴλιον ἐκάλεcε, τῷ δὲ Διὶ cηµεῖον εὐξάµενοc αὐτῷ τι φανῆναι, µεθ᾿ ἡµέραν τὸ διιπετὲc παλλάδιον πρὸ τῆc cκηνῆc κείµενον ἐθεάcατο. ἦν δὲ τῷ µεγέθει τρίπηχυ, τοῖc δὲ ποcὶ cυµβεβηκόc, καὶ τῇ µὲν δεξιᾷ δόρυ διηρµένον ἔχον τῇ δὲ ἑτέρᾳ ἠλακάτην καὶ ἄτρακτον. III, 144-145 [ἱcτορία δὲ ἡ περὶ τοῦ παλλαδίου τοιάδε φέρεται• φαcὶ γεννηθεῖcαν τὴν Ἀθηνᾶν παρὰ Τρίτωνι τρέφεcθαι, ᾧ θυγάτηρ ἦν Παλλάc• ἀµφοτέραc δὲ ἀcκούcαc τὰ κατὰ πόλεµον εἰc φιλονεικίαν ποτὲ προελθεῖν. µελλούcηc δὲ πλήττειν τῆc Παλλάδοc τὸν Δία φοβηθέντα τὴν αἰγίδα προτεῖναι, τὴν δὲ εὐλαβηθεῖcαν ἀναβλέψαι, καὶ οὕτωc ὑπὸ τῆc Ἀθηνᾶc τρωθεῖcαν πεcεῖν. Ἀθηνᾶν δὲ περίλυπον ἐπ᾿ αὐτῇ γενοµένην, ξόανον ἐκείνηc ὅµοιον καταcκευάcαι, καὶ περιθεῖναι τοῖc cτέρνοιc ἣν ἔδειcεν αἰγίδα, καὶ τιµᾶν ἱδρυcαµένην παρὰ τῷ Διί. ὕcτερον δὲ Ἠλέκτραc κατὰ τὴν φθορὰν τούτῳ προc-φυγούcηc, Δία ῥῖψαι µετ᾿ αὐτῆc/ ᾽Ατης 38 καὶ τὸ παλλάδιον εἰc τὴν Ἰλιάδα χώραν, Ἶλον δὲ τούτῳ ναὸν καταcκευάcαντα τιµᾶν. καὶ περὶ µὲν τοῦ παλλαδίου ταῦτα λέγεται.] III, 144-145. Dionysios of Halicarnassos, I, 68 Εἰς πόλιν ἣν κτίζῃσθα θεοῖς σέβας ἄφθιτον αἰεὶ θεῖναι, καὶ φυλακαῖς τε σέβειν θυσίαις τε χοροῖς τε• εὖτ᾽ ἂν γὰρ τάδε σεµνὰ καθ᾽ ὑµετέραν χθόνα µίµνῃ δῶρα Διὸς ἀλόχῳ σέθεν, ἥδε πόλις σοι ἔσται ἀπόρθητος τὸν ἀεὶ χρόνον ἤµατα πάντα. Minor parallels on Ilos: Ἐν Ἴλίῳ τοῦ ναοῦ τῆς Ἀθηνᾶς ἐµπρησθέντος, προσδραµὼν Ἶλος τὸ διοπετὲς ἥρπασε παλλάδιον ἐτυφλώθη. οὐ γάρ ἐξῆν ὑπ ̓ ἀνδρὸς βλέπεσθαι, ὔστερον δ̓ ̓ ἐξιλασάµενοc ἀνέβλεψεν (Derkyllos and Aristides of Miletos) Procl. Chrest. 206 (Iliadis Parvae enarratio)… Όδυσσεύς τε αἰκισάµενος ἑαυτὸν κατάσκοπος εἰς Ἴλιον παραγίνεται, καὶ ἀναγνωρίσθεὶς ὺφ` Ὲλένης περὶ τῆς ἁλώσεως τῆς πόλεως συντίθεται κτείνας τέ τινας τῶν Τρώων ἐπὶ τὰς ναῦς ἀφικνεῖται. Καὶ µετὰ ταῦτα σὺν Διοµήδει τὸ παλλάδιον έκκοµίζει. (Iliupersidos enarratio) Μενέλαος δὲ ἀνευρὼν Ἑλένην ἐπὶ τᾶς ναῦς κατάγει, Δηίφοβον φονεύσας. Κασσάνδραν δὲ Αἴας ὁ Ἰλέως πρὸς βίαν ἀποσπῶν συνεφέλκεται τὸ τῆς ᾽Αθηνᾶς ξόανον. ἐφ᾽ ᾧ παροξυνθέντες οἱ Ἓλληνες καταλεῦσαι βοθλεύονται τὸν Αἴαντα. ὁ δὲ ἐπὶ τὸν τῆς ᾽Αθηνᾶς καταφέυγει καὶ διασῴζεται ἐκ τοῦ ἐπικειµένου κινδύνου. Quintus of Smyrna, 10, 350-360 οὐδὲ γὰρ οὐδ`θεῶν τις ἀπειρέσιον χαλεπήνας ἔσθενεν ὄλβιον ἄστυ διαπραθέειν Πριἀµοιο ἀθανάτης ἔµπροσθεν ἀκηδέος ἐµβεβαυίης ἁρπάξας ἐθέλουσαν ἐύφρονα Τριτογένειαν / ἢ τ῎ ἔρυµα πτὀλιός τε καὶ αὐτῶν ἔπλετο Τρώων•); Virgil, AEn. 2, 171-179 Nec dubiis ea signa dedit Tritonia monstris. Vix positum castris simulacrum, arsere coruscae luminibus flammae arrectis, salsusque per artus sudor iit, terque ipsa solo --mirabile dictu -emicuit, parmamque ferens hastamque trementem. 'Extemplo temptanda fuga canit aequora Calchas, nec posse Argolicis exscindi Pergama telis, omina ni repetant Argis, numenque reducant, quod pelago et curuis secum auexere carinis.(cf. comm. by Servius) Dictys Cretensis 5.5-7: focus on Antenor and Theano as traitors Praeterea cognoscunt ab Antenore editum quondam oraculum Trojanis, maximo exitio civitati fore, si Palladium, quod in templo Minervae esset, extra moenia tolleretur : namque id antiquissimum signurn caelo sublapsum, qua tempestate Ilus templum Minervae extruens, prope summum fastigii pervenerat, ibique inter opera, quum necdum tegumen superpositum esset, sedem sibi occupavisse, idque signum ligno fabrefactum esse. Hortantibus dein nostris, uti secum ad ea omnia eniteretur, facturum se quae cuperent, reepondit. Atque his praedicit, publice se in consilio super qualitate eorum quae postulaturi essent, exertius disserturum, scilicet ne qua suspicio sui apud Barbaros oriretur. Ita composito negotio, cum luce simul Antenor ac reliqui proceres ad Priamum vadunt; nostri ad naves redeunt. 6 Dein ubi justa pueris farca sunt, post diem tertium Idaeus supradictos duces accitum venit : queis praesentibus Lampus, caeterique quorum consilium praevalebat, multa disserere, atque docere ea quae antea gesta essent temere et inconsulte, non per se, quippe qui contempti despectique a regulis, arbitrio alieno agerent. Caeterum quod arma adversus Graecos tulissent, non sponte factum : namque qui sub imperio alieno agerent, expectandum his, atque exsequendum esse nutum ejus qui teneat. Ob quae dignum esse Graecos data venia consulere eis, qui semper auctores pacis fuerint : caeterum a Trojanis ob maleconsulta satis poenarum exactum. Dein multo hinc atque inde habito sermone, ad postremum de modo praemiorum agi coeptum. Tum Diomedes quinque millia talentorum auri, ac totidem argenti optat : praeterea tritici centena millia, eaque per annos decem. Tum silentio habito a cunctis, Antenor, non Graecorum more agere eos adversum se ait, sed barbaro : namque quod impossibilia postularent, palam fieri, praetextu pacis bellum eos instruere : caeterum auri tantum atque argenti ne tum quidem priusquam in auxilia conducta dilaceraretur, civitati fuisse. Quod si permanere in eadem avaritia vellent, superesse Trojania, uti clausis portis, incensisque intus deorum aedificiis, ad postremum idem sibi cum patria exitium peterent. Contra Diomedes: « Non civitatem vestram consideratum Argis venimus, verum adversum vos dimicaturi. Quodcirca sive etiam nunc bellare in animo est, parati Graeci : sive, ut ais, igni dabitis Ilium, non prohibebimus : quippe Graecia affectis injuria, ulcisci hostes suos finis est. »

	Annex: ancient texts
	[Apollodorus], Id., II, 66
	τὸ διιπετὲς παλλάδιον
	Plutarque,

Λέγουσι γοῦν ὧδε• Χρύσην τὴν Πάλλαντος θυγατέρα γηµαµένην Δαρδάνῳ φέρνας ἐπενέγκασθαι δωρεὰς ᾽Αθηνᾶς τά τε Παλλάδια καὶ τὰ ἱερὰ τῶν µαγαλῶν θεῶν διδαχθεῖσαν αὐτῶν τᾶς τελετάς• I, 69,2 ἄραντα ἐκ τῶν ἀδύτων τά τε ἱερὰ τῶν µεγάλων θεῶν καὶ ὅπερ ἔτι περιῆν Παλλάδιον (θάτερον γὰρ Ὀδυσσέα καῖ Διοµήδην νυκτὸς φασιν εἰς Ἴλιον ἀφικοµἐνους) (fr. of Arctinos) I, 69.3 Άρκτῖνος δὲ φησιν ὑπὸ Διὸς δοθῆναι Δαρδάνῳ Παλλάδιον ἓν καὶ εἶναι τοῦτο ἐν Ἴλίῳ τέως ἡ πόλις ἡλίσκετο I, 69.4 καὶ τὸ µυθευόµενον Παλλάδιον. ὅ φασι τὰς ἱερὰς φυλἀττειν παρθένους ἐν ναῷ κείµενον Ἑστιᾶς.

The iconographical part of the research due to my colleague Maria Paola Castiglioni was orally presented with mine in Greece at a conference on the Epic Cycle. Unhappily she did not have time for revising her text, so that I remain sole responsible for this paper, including the errors that may occur. My warm thanks also go to Javier Martinez who asked me to take part to the second edition of the book and to Stephen Rojcewicz who revises my English with a kind akribeia.

Let us refer to Milman Parry's definition of the formula, and recall that Achilles may be called 'swift footed' (gr. Πόδας ὠκύς. ποδωκύς) even when seated (Graziosi 000000).

End of 1.67: ἐγὼ δὲ ὅσα µὲν ὁρᾶν ἅπασιν οὐ θέµις οὔτε παρὰ τῶν ὁρώντων ἀκούειν οὔτε ἀναγράϕειν οἴοµαι δεῖν, νεµεσῶ δὲ καὶ τοῖς ἄλλοις, ὅσοι πλείω τῶν συγχωρουµένων ὑπὸ νόµου ζητεῖν ἢ γινώσκειν ἀξιοῦσιν. Beginning of 1.67: ῝Α δὲ αὐτός τε ἰδὼν ἐπίσταµαι καὶ δέος οὐδὲν ἀποκωλύει µε περὶ αὐτῶν γράϕειν τοιάδε ἐστί•

Dardanos was the city's name for Dionysios, whereas the Iliad mentioned rather Dardania, higher in the Ida's slopes than Troy.

It is actually a prosaic transposition of the oracular text mentioned above.

Of course, since the word Palladion is not there, a research in the TLG missed this occurrence.

350)). An Attic cup (470-460) and a Panathenaic amphora (about 420) depict only one palladion in accordance with the Little Iliad's version.

Argos, Sparta, Thebes, Athens, Amphissa, Rome, Siris, Luceria, Lavinium.

Three mythological versions at least may be distinguished : -wedding gift to Dardanos' wife by Athena (who then may have fabbricated it previously) -fallen before Dardanos' door -fallen before Ilos' door Dionysios Halicarnassensis mentions both versions relating to Dardanos, [Apollodorus] the version relating to Ilos and Athena's fabrication.

His frequent courtships imply, however, that he sometimes descends to earth and takes a human form -Amphitryon's for example-to seduce women such as Alcmene;

Which could explain one of the traditional epithets of the Goddess, Tritogeneia, at least in popular etymology, as it often occurs in Antiquity.


Pierre Carles

Stéphane Popinet

Viscous non-linear theory of Richtmyer-Meshkov Instability

We propose a quantitative prediction of the effect of viscosity on the weakly non-linear impulsive Richtmyer-Meshkov instability between two fluids of arbitrary densities and viscosities. This theory is based on an asymptotic analysis of the Navier-Stokes equations using singular perturbation techniques. The law obtained for the interface deformation does not agree with former theoretical predictions of the effect of viscosity (Mikaelian, PRE 47, p.375 (1993)), but compares very well with direct numerical simulations we performed using a front-tracking code developed in our laboratory.

When a plane shock wave travels across an interface separating two fluids of different densities, all perturbations on the interface grow with time and a strong mixing of the two fluids ensues. This instability, named the Richtmyer-Meshkov (RM) instability (after it was theoretically predicted by Richtmyer [1] and experimentally observed by Meshkov [2]), appears in many different technical and scientific domains, including supersonic combustion, inertial confinement fusion and supernovae explosions. Very recently, a suggestion was made by Gueyffier and Zaleski [3,4] that the RM instability could also play an important role in impact phenomena between liquids and solids or liquids and liquids.

From a theoretical point of view, the RM instability can be approached in two ways: by linearizing Euler's equations for small interface perturbations around a base solution corresponding to the 1D propagation of a shock through a flat interface; or by assuming the fluids are incompressible and representing the passage of the shock by an impulsive acceleration perpendicular to the interface. The first approach has proved very efficient in predicting the maximum rate of growth of the perturbations for different initial conditions [1,[5][6][7], but almost impossible to extend to the non-linear regime. On the other hand, the impulsive model for incompressible fluids is more appropriate for studying weak nonlinear effects [8,9] and multi-mode coupling [10], but usually over-estimates the maximum growth rate. Recently, Zhang and Sohn [11] bridged the gap between the two approaches by using the linear compressible solution for describing the short-time evolution and a matched impulsive incompressible model for later times. Several authors in the past thirty years have discussed the limits of the impulsive model, often proposing methods for extending its validity to a larger range of parameters [5,7,[12][13][14][15].

In all these approaches however, viscous effects have been systematically disregarded, with the exception of a contribution by Mikaelian which we will discuss later [16]. Although viscous effects are often assumed to be negligible in most of the above-mentioned cases, this assumption is seldom based on quantitative arguments. The aim of this letter is to present a new, viscous, weakly non-linear theory of the RM instability enabling quantitative predictions of the intensity of viscous effects for two fluids of arbitrary densities and viscosi-ties. These predictions are compared to direct numerical simulations conducted with a full Navier-Stokes front-tracking code, and the excellent agreement which is obtained confirms the validity of the theory.

The theory is based on an asymptotic analysis of the Navier-Stokes equations written for two infinite incompressible viscous fluids separated by an interface with no surface tension (the effect of surface tension can be included in a very straightforward manner and will be discussed in more detail in a forthcoming paper). The fluids (noted 1 and 2) are initially at rest, the interface is slightly perturbed and an impulsive vertical acceleration is imposed in the 1-2 direction. Without viscosity, a discontinuity in the tangential velocity appears at the interface which subsequently becomes a vortex sheet. If viscosity is present, vorticity diffuses from the interface into the fluids and the growth rate of the instability is affected.

In the present work, the viscous effects are taken into account in two boundary layers located on each side of the interface, in which equations analogous to Prandtl equations are treated. Far from the interface, inviscid equations are solved and the technique of Matched Asymptotic Expansions [START_REF] Kevorkian | Perturbation methods in applied mathematics[END_REF] is used to link the two systems in each fluid. Such a boundarylayer treatment imposes a restriction on the local Reynolds numbers in each fluid, which is always satisfied in experimental cases of practical interest (even in the case of liquid-liquid interfaces as in Jacobs and Sheeley's experiments [START_REF] Jacobs | [END_REF]).

Let us suppose that the perturbed interface is initially defined by the equation:

z = η (x, t = 0) = η 0 cos (kx)
(the x-axis is in the plane of the non-perturbed interface, the z-axis perpendicular to it and oriented in the opposite direction to that of the impulsive acceleration). The impulsive acceleration is defined as g (t) = ∆V δ (t). In order to make an asymptotic expansion of the equations of the problem, they are first written in non-dimensional form, using the following normalizations (the superscript "(i)" stands for fluid 1 or 2):

x = xk, z = zk, t = ∆V k t, η = η η 0 , u (i) = u (i) ∆V , w (i) = w (i) ∆V , P (i) = P (i) (ρ (1) + ρ (2) ) ∆V 2 .
The non-dimensional equations then read as follows, omitting the primes (subscripts stand for partial derivatives):

u (i) x + w (i) z = 0, u (i) t + u (i) u (i) x + w (i) u (i) z = - ρ (1) + ρ (2) ρ (i) P (i) x + 1 Re (i) u (i) xx + u (i) zz , w (i) t + w (i) w (i) x + w (i) w (i) z = - ρ (1) + ρ (2) ρ (i) P (i) z + 1 Re (i) w (i) xx + w (i) zz -δ (t)
with

(i) Re = ∆V ρ (i) µ (i) k ,
where µ (i) and ρ (i) are the viscosity and the density of fluid i. Far from the interface, the velocity vanishes and the pressure gradient becomes purely hydrostatic. At the interface (i.e. for z = kη 0 η ), the following boundary conditions hold: 2) , w (1) = w (2) , u (1) z + w (1) x = µ (2) µ (1) u (2) z + w (2) x , P (1) -2 ρ (1) ρ (1) + ρ (2) 1 Re (1) w (1) z = P (2) -2

u (1) = u (
ρ (2) ρ (1) + ρ (2)
1 Re (2) w (2) z .

Finally, the condition that the interface be a material surface reads (again for z = kη 0 η ):

η t = 1 kη 0 w (i) -η x u (i) .
The usual condition of small initial perturbations is applied, which translates into: ε = kη 0 << 1. In terms of the non-dimensional z variable, the thickness of the deformed interface is thus of order ε. In order to limit the viscous effects to the vicinity of the interface, the following condition is imposed:

Re (i) = O (ε -2 ).
A detailed asymptotic analysis shows that such a choice, among all the possible scalings for large Re (i) , leads to the most descriptive expansion. This conclusion is easy to reach on purely physical grounds: this choice indeed leads indeed to a boundary layer thickness of the same order as the interface deformation (i.e. ε). For larger values of Re (i) , the boundary layer is smaller than the interface deformation and the curvature of the interface is neglected in the calculation of the viscous effects. For smaller values of Re (i) on the contrary, the system in this thicker boundary layer degenerates, being unable to take into account the continuity of viscous stresses at the smaller scale of the interface. Hence, values of Re (i) smaller or larger than ε -2 , provided they remain larger than 1, lead to systems which are particular cases of the expansions obtained for Re (i) of order ε -2 . Note that the condition that the Reynolds numbers Re (i) be large compared to one is not a drastic one. Indeed, the velocity used to define these Reynolds numbers is not the average velocity of the fluid but the much larger velocity ∆V used in the definition of the impulsive acceleration g (t). These "impulsive" Reynolds numbers are always very large in usual experiments (like in Ref. [START_REF] Jacobs | [END_REF] or [19]).

Having defined all the non-dimensional quantities involved and their scaling with ε, it is now possible to construct asymptotic series for all the variables of the problem (i.e. η, u (i) , w (i) , P (i) ). The external systems (far from the interface) are obtained by making ε tend to zero for x and z fixed. The boundary layer systems are obtained by making ε tend to zero for x and z fixed, with z the boundary layer space variable, defined as: z = z /ε. The matching between external and boundary layers systems is obtained using a standard Matched Asymptotic Expansions technique. The pressure and velocity fields can be found by variable separation, solved in the physical space for external systems and in the Laplace space for the boundary layers. The inverse Laplace transforms of these solutions can be obtained analytically in the case with no surface tension and for the first two orders of expansion. This rather involved calculation will be detailed in a paper in preparation.

In this letter, we only present the second-order expansion in ε of the interface equation, obtained through this process:

η (x , t ) = (1 + At ) cos (x ) -ε 8A 3 √ π f ρ (i) , µ (i) t
number, equal to ρ (1) -ρ (2) / ρ (1) + ρ (2) . When translated into dimensional form, this expansion gives the following viscous weakly non-linear law of interface deformation, the main result of this letter:

η η 0 (x, t) = (1 + A ∆V k t) cos (kx) - 16 3 √ π ρ (1) µ (1) ρ (2) µ (2) (ρ (1) + ρ (2) ) ρ (1) µ (1) + ρ (2) µ (2) A ∆V k 2 t 3 2 cos (kx) (1)
- 1 2 η 0 A 3 ∆V 2 k 3 t 2 cos (2kx) .
The first term in the sum is the usual growth of the interface perturbation in the linear impulsive model; the second term represents the viscous attenuation of this growth; the third term is the first non-linear coupling of the perturbation with itself. Note that this third term is identical to the one obtained by Zhang and Sohn [8] or Berning and Rubenchik [10] in the non-viscous case. This identical non-linear term, apart from being an indirect validation of the whole calculation, shows that in the majority of cases (i.e. when the impulsive Reynolds numbers are large), viscous attenuation mainly affects the first deformation mode.

A third-order expansion in ε would in turn give access to the second non-linear coupling of the perturbation with itself (with an x-dependence under the form of cos(kx) and cos(3kx)), together with the term representing the viscous attenuation of the first non-linear mode (in cos (2kx)). However, owing to the complexity of the calculation already at second order, such an expansion seems out of reach. Note that beside the interface deformation, the present calculation gives also access to the full velocity and pressure fields in the two fluids.

This can be a very interesting information, for instance to study the evolution in time of the vorticity initially deposited at the interface by the shock (this point will also be discussed in the forthcoming article mentioned above).

In Ref. [16], Mikaelian proposed an expression for the linear RM instability between two viscous fluids. His method was based on moment equations applied to a non-viscous approximation of the vertical velocity, and lead to the following law of interface deformation:

η η 0 (x, t) = 1 + A ∆V 1 -e -2k 2 νt 2kν cos (kx) with ν = µ (1) + µ (2) ρ (1) + ρ (2) . (2)
The expansion of this law for small times yields:

η η 0 (x, t) = (1 + A ∆V k t) cos (kx) -A ∆V k 3 νt 2 cos (kx) ,
which should be compared to the first two terms of expression (1). The difference is both qualitative and quantitative: the power law in time is different, and for most cases the two expressions for the viscous correction differ by several orders of magnitude.

In order to check the validity of our theory, we have made a number of numerical simulations using a full Navier-Stokes incompressible front-tracking code developed by Popinet and Zaleski in recent years [20]. This code has been shown to give very accurate results, particularly in the case of the viscous damping of capillary waves. In the present study, two periodic layers of incompressible viscous fluids separated by a perturbed interface are initially superposed between two rigid boundaries. The distance between the boundaries is equal to the wavelength of the perturbation (using larger simulation domains does not change the results significantly, which shows that the effect of confinement is negligible).

The domain is discretized using a regular cartesian grid (256 2 cells) and an adaptive timestepping procedure is used to resolve the very short initial time scales. A strong acceleration is imposed for a very short time, and the response of the fluids is simulated. A Fourier transform of the shape of the interface is then calculated, and the first two terms (mode one and two) are compared with formula (1). Note that in order to increase precision in the simulation of the interface evolution, the dynamic viscosities of the fluids have been chosen equal (see Ref. [20,21]). The densities however are different, which leads to different kinematic viscosities.

The results obtained for a density ratio of 1/4 in an unstable configuration (ρ (1) > ρ (2) )

and for several viscosities are presented in figures 1 and 2. Simulations with different density ratios have been performed. Similar results are obtained but are not detailed here. The first Fourier component (associated with cos(kx)) minus the linear growth (1 + A ∆V k t) is represented in figure 1 for different viscosities. The agreement between the numerical results and the predictions of Eq. ( 1) is very good. The viscous correction predicted by Mikaelian (i.e. Eq. ( 2) minus the linear growth (1 + A ∆V k t)) is also illustrated in figure 1 for the same series of Reynolds numbers. This comparison shows unambiguously that Eq. ( 2) is inappropriate.

The second Fourier mode (associated with cos(2kx)) is represented in figure 2. As viscosity decreases (large Reynolds numbers), the numerical result converges to the analytical one. The discrepancy between the theoretical and numerical curves can therefore clearly be attributed to the viscous attenuation of the growth of the second mode, not taken into account in the present theory. Note that this comparison is, to our knowledge, the first numerical confirmation of the validity of the first non-linear correction as calculated in Ref.

[8] and [10] or in the present work.

In conclusion, we propose a quantitative prediction of the influence of viscosity on the RM instability, under the form of the interface deformation law given in Eq. ( 1). This theoretical law is in excellent agreement with the results of direct numerical simulations. Of course, the proposed asymptotic expansion is secular in time, so that it eventually fails after the interface has penetrated too deep into the non-linear regime. But the present theory remains nonetheless an important tool for two purposes: to estimate the influence of viscous effects on the RM instability, and in particular to provide quantitative arguments about the relevance of non-viscous theories; to serve as a benchmark for full Navier-Stokes numerical codes for two-phase flows, for which the Fourier analysis of the interface deformation obtained in the impulsive RM instability is a stringent test of precision.

FIGURESFIG. 1 .Figure 2

 12 FIGURESFIG.1. Viscous correction to the linear growth based on Eq. (1), Eq. (2) (Mikaelian[16]) and numerical simulations for several Reynolds numbers.

cos (x ) -ε 1 2 A

t 2 cos (x ) ,where f is a function of the densities and the viscosities of the fluids, and A the Atwood

The authors wish to acknowledge Dr. Denis Gueyffier for giving the initial motivation for this study.