Samuel Trainor

Mon coeur est vide; mon esprit est contraint : je joue le premier personnage de France; mais ce n'est qu'un personnage. Je ne vis que d'une vie empruntée. 1 (Voltaire, Dialogue entre M me de Maintenon et Mlle de Lenclos 1751, 1754) Froissart's forge C'est (qu')en forgeant qu'on devient forgeron. In the bureau de change of the bilingual dictionary, this finely wrought French epigram is traded for the English coinage: 'practice makes perfect.' There's a steep commission to be paid on the transaction, though. You lose the sturdy antitheoretical insistence of the French: 'it is by smithying [alone] that one becomes a blacksmith'… fine words butter no parsnips. That said, the martelato motto hammers home its point: 'practice makes the practitioner.' This is truism as tautology. By the same token, we should be wary of its final word. Forgeron is a faux ami. A mistranslation will impress itself upon the unannealed mind of the apprentice reader: 'it is by forging that one becomes a forger.' This paper will take up that misstruck metaphrase to test the mettle of an infamous fictitious history… so infamous, in fact, that it circulated for a century, in the culture that defined historical modernity, as the archetype of counterfeit historiography. The question to be posed, therefore, is this: 'is it by forging history that one becomes a historian?'

Jean Froissart lost little sleep over the idea: Je me suis de nouvel réveillé et entré dedans ma forge pour ouvrer et forger en la haute et noble matière de laquelle du temps passé (Chroniques Book 4 p.1, around 1400) This ductile rendering of the craft of the medieval chronicler is unapologetically a matter of synthetic manufacture. The past is just material to his art. It is important to him that this material be 'noble' (by which he means 'permanent' as much as 'valuable': he casts himself as goldsmith, rather than a whittler of firewood… and we can take the master craftsman at his word that this 'nobility' is something like the stuff of truth). Its value notwithstanding, though, this high and noble substance is still only an artist's medium. The history is the art. It may be just a metaphor, but it's one that he's content to live by, even if it means him flirting with some deconstructive ambiguities. If, that is, that noble stuff -le temps passé -is simply 'the past tense', the story of the deposition of Richard II might become as insubstantial as an exercice de style: no more significant an incident than a young man on a Paris bus sticking his long neck out to accuse another passenger of pushing him around. 2 Rather than goldsmith to a King -subject to the assay of Archimedes, that gilt-edged fable of empiricism -Froissart might turn out to be no more than tailor to a naked emperor.

In any case, the distinction made by Aristotle between history and poetry (the recounting of 'the [singular] thing that has been' versus the fashioning of 'the [universal] thing that might be' -Poetics Book 9) is implicitly refused by Froissart's implicit concept of historiopoesis. The story of how this classical dichotomy is reasserted by Enlightenment historiography, 3 and in turn resisted by a proponent of Baroque pastiche, is the intrigue of this essay. It plays out against the backdrop of a scramble in the eighteenth century French public sphere 4 to forge a new intellectual epoch (l'age des lumières: 'the Enlightenment'… though the view that this idea might be self-consciously emergent from the era it describes is itself historically problematic, see note 15). Crucial to this intellectual transformation is (or was) a revisionist attempt to posit the origins of the supposed new era in the cultural transactions of the previous century: the so-called 'Grand Sciècle' of Louis XIV. Not only is the story of the reign of Louis XIV fundamental to the metanarrative of a nascent age of reason, but the neo-classical methodology of the objective historical perspective (precisely analogous to the classical perspective of renaissance painting, which had undergone its own Baroque refractions in that tumultuous century) is also key to the construction of the new social-scientific discipline: l'histoire. It marks a definitive break from the medieval notion of history as poesis, as represented by Froissart's forge. Gone, almost by definition, is all the imaginative and ideological jiggery-pokery that might go on under cover of such a sooty -or an inky -ouvroir.

Madame of the Moment

The arbitrary object of this debate is not Louis XIV himself, but his second wife: Françoise d'Aubigné, the Marquise de Maintenon. A figure with a seemingly insurmountable moment of historical inertia, this remarkable woman became the holy grail of eighteenth century histories of the Grand Siècle. The displacement of the centre of courtly revolution, from the Sun-King himself to his mercurial companion, is a quasi-Copernican realignment of the historical point de vue 6 which owes as much to the aptitude of her life's story to the burgeoning genre of prose fiction as it does to the suspicion that she had become the power behind the throne. She was a woman -one who had earned her pre-eminent position in history with her womanly wiles… or so the story goes -and this simple fact would make her all the more desirable as a protagonist... as a romantic heroine. Think Becky Sharp meets Princess Diana 7 and you might at least get an idea of the scale (the moment and the momentum 8 ) of this problematic personnage. Whatever the analogues -and none of them match up -no summary of her life can fail to sound like the plot of an eighteenth century novel. No point in bucking the trend. Here is the cover blurb to the paperback edition (2009) of Veronica Buckley's (2008) biography:

The rags-to-riches story of Françoise d'Aubigné, wife of the Sun-King of France Françoise d'Aubigné, born in a bleak provincial prison, her father a condemned murderer and traitor to the state, rose from the depths of poverty to life at the vortex of power at Versailles. Married at fifteen to a tragically disfigured and scandalously popular poet, in his salon Françoise encountered all the brilliant characters of the seventeenth century's glitterati. After her husband's death, she led the life of a merry widow in the colourful Marais quarter of Paris, before becoming governess to the King's growing brood of royal bâtards. This is the extraordinary story of one woman's daring journey from beggar-girl, West Indian colonist and salonnière to royal mistress and thence, in secret, to the compromised position of Louis' uncrowned Queen.

(Cover blurb: Madame de Maintenon, the Secret Wife of Louis XIV, Veronica Buckley, London, Bloomsbury 2009 [paperback ed.])

Such an incorrigibly bubbly apéritif necessarily obscures the fact that copies of the first hardback edition were sold including one of the most embarrassing erratum slips in the history of biography. It confessed that the two glaring references made in the book to a secret diary written by the King and sealed, according to his deathbed instructions, for a period of 150 years, were the result of author and editor being duped by the pastiche Journal secret de Louis XIV published in 1998 by François Bluche. Whilst Bluche's text does not explicitly reveal itself to be pastiche (this being the nail in the coffin of pastiche), its own cover blurb makes a claim of authenticity which French readers (sensitized to this sort of thing by a long tradition of apocrypha supposedly emanating from the Sun-King's court) 9 are unlikely to mistake as anything but tongue in cheek:

Ce Journal secret de Louis XIV présente de fait tous les caractères de l'authenticité : événements vérifiés, dates exactes, analyses politiques et psychologiques vraisemblables. Tous les faits importants d'un grand règne y sont racontés ou évoqués : de la disgrâce de Fouquet à la révocation de l'Édit de Nantes, du mystère du Masque de fer aux secrets d'alcôve... toute l'histoire du Grand Siècle vue par celui qui le marqua de sa stature : le Roi-Soleil en personne. 10 (Cover blurb: Le journal secret de Louis XIV, François Bluche, Paris, Editions du Rocher, 1998) For an opportunistic English reader to have missed this irony, and to have been blind to such an evident pastiche, might be dreadfully embarrassing, but it's far from unforgivable. The unsuspecting reader, trawling for a juicy bodice ripper in the annals of French history, cannot fail to drag up M me de Maintenon. In doing so, she revisits a tale that is the epitome of the unreliable history. There is seemingly no way that the 'real' story of M me de Maintenon can be extracted from the pastiche. It is as if she were, herself, a pastiche (a product of poesis). A brief explanation of why is provided below (only one amongst many), but I will limit myself here to a note of Jacques Derrida's that puts the case as firmly as possible (given the time [un]available to him, and to me, in an academic conference):

Madame de Maintenon ne me paraît pas seulement exemplaire parce qu'elle pose la question du don du temps -et du reste -, depuis la place d'une femme et d'une grande dame. Celle qui joua auprès de Louis XIV le rôle d'une «;sultane de conscience;» fut à la fois, configuration rarement fortuite, un horsla-loi et la figure même de la loi. Avant de devenir, après la mort de la reine, l'épouse morganatique du roi (ainsi exlue du nom et des droits nobiliaires -et le mot «;morganatique;» dit quelque chose du don, du don de l'origine : il vient du bas latin morganegiba, don du matin), elle avait ramené le Roi Soleil à ses devoirs d'époux (en l'éloignant de M me de Montespan dont elle avait été la protégée) et de roi catholique (en rappelant la cour à l'austérité, en encourageant la persécution des protesants -bien qu'elle eût été élevée dans le calvinisme -, et en apportant son soutien à la révocation de l'édit de Nantes). Celle qui se donna tant de mal avec ce qu'il fallait prendre et donner, avec la loi, avec le nom du roi, avec la légitimité en général, fut aussi la gouvernante de bâtards royaux, promotion qu'elle dut sant doute à la protection de M me de Montespan. Arrêtons-nous là où il eût fallu commencer : alors qu'elle était enfant, elle connut l'exil en Martinique et Constant, son père, fut arrêté comme faux-monnayeur. Tout dans cette vie paraît marqué au coin le plus austère, le plus rigoureux, le plus authentique de la fausse monnaie. 11

(Jacques Derrida, Donner le temps 1. Derrida, however, also omits to tell us that even the 'authentic' letters of M me de Maintenonsuch that they are retrospectively identifiable -cannot (despite their candour, their piety, their limpid prose) be trusted to be 'sincere' (or 'authentic', in Lionel Trilling's terms: see note 22 Sire).

The overriding characteristic of M me de Maintenon's writings, in comparison for example with those of M me de Sévigné, is a remarkable historical self-consciousness. The effect seems to be general enough to convince the modern reader that this goes beyond the inevitable taint of hindsight that the fakes of La Beaumelle's collection have left upon their neighbours. 13 Simply put, M me de Maintenon, as she comes down to us, is the self-conscious historical personnage par excellence, a writer who always has one eye on posterity (while posterity has one eye on her posterior). She implicitly identifies the neo-classical historical point of view (pre-empting its assertion in the critical debates about her putative writings) as a libidinous male gaze intent upon submitting her personnage (like her body) to a fixed (one-track) perspective -a would-be penetrative gaze precisely analogous to those of the men in the court, and of the King himself. Subjected to this gaze, she does what she has always done: she demures… with all the ironies, flirtations and manipulations that donning such an ostentatious mode of modesty entails. That, at least, is how she looks to us from here:

Votre retenue… au lieu de me lasser, m'a aiguisé l'appetit. Je vois bien dans cette pruderie feinte l'effet de votre conscience de l'amour. Above all, this makes M me de Maintenon ungraspable as a personality: physically (as a sexual object not to be touched), historically (as a problematic site of regression of the historical perspective), and temporally (as a prosaic consciousness with a multiplicitous -very much a textual -tensuality that has no stable 'now'… a writer whose elegantly folded projections, recollections, speculations and suggestions necessarily undermine the fixed point of time and space inscribed at the start of every letter). In short, M me de Maintenon can no more be 'grasped' (saisie) or 'maintained' ('held' -tenue -'in the hand' -à la main) than can the concept of 'now' in the French language. This is the last thing Derrida is (not) going to tell us: M me de Maintenon is, homophonically, M me de Maintenant (Madame Now)… a pun that all too easily gets out of hand:

Je pense que chaqune de mes amies correspond à une réalité de mon coeur ou de ma tête : la duchesse [de Fontagnes] à ma faiblesse, Ath In the mid eighteenth century, when this story of the Grand Siècle was being brought into focus (mise au point), the man of the moment was, with only the faintest shadow of a doubt, Voltaire. 15 It was in order to protect this status, and the pre-eminence of his Siècle de Louis XIV as the text which defined an era, 16 that Voltaire staged his demolition of La Beaumelle's 'edition' of M me de Maintenon's 'memoirs' and letters. Théophile Lavalée (1865 vol.1 p.XVI) suggests that Voltaire had been awaiting the publication of this book with dread, fearing that the priveleged access to the archives of Louis Racine (son of Jean Racine, poet and official historiographer to Louis XIV) and those of Saint-Cyr -which La Beaumelle had seemingly tricked his way into procuring -would mean that the young protestant pretender's text would discredit his own powerfully conceived, but patchily researched account. 17 The relief and relish with which he goes about his critical assassination of La Beaumelle is almost palpable. In two sections of his Honnêtetés Littéraires (1767) dedicated to this exiled upstart, and most vehemently in the appended 'Lettre à l'auteur...', Voltaire really lets rip. He calls La Beaumelle a misérable calomniateur and 'a little Huguenot raised to be a preacher, who has never seen anything; who has spoken as if he had seen it all; who writes in a style as audacious as it is impertinent'. Most critically (and most hypocritically) of all he accuses La Beaumelle of being a romancier (a 'novelist'), and as a coup de grâce: On voit à chaque page un homme... qui ne songe qu'à écrire un roman.

The mutual exclusivity of history and fiction is glaringly implied. However, this is not, as one might imagine, principally a question of proscribing the manufacture of events or sources. Beyond ridiculing the factual incoherences it turns up, Voltaire has little to say about the fact of La Beaumelle's counterfeiting itself. Instead, the nitty gritty of his critique denounces his rival's application of fictional stylistics and (for want of a better term) 'poetic license' to an object of study which, according to moral and intellectual criterea he is fixing here himself (though seemingly not for himself: see epigraph), cannot be treated in this way. Particular opprobrium is reserved for those moments in which these techniques imply an intimate knowledge of characters' private or inner lives, and a license to speak with authority about them, that are chronologically and socially transgressive. In particular he seeks to expunge the over-reaching assumption of the priveleged 'points of view' of novelistic narrative: « … dans un déshabillé léger, [elle] s'était jetée dans un fauteuil; là elle pensait à loisir à son amant ; souvent le jour la retrouvoit assise sur une chaise, accoudée sur une table, l'oeil fixe dans l'extase de l'amour. » Hé, mon ami! l'as-tu vue dans ce déshabillé léger? L'as-tu vue accoudée sur cette The central rhetorical question of Voltaire's critique splits the debate right down the middle: 'Is it permissible to write history in this way?', might just as easily mean 'Is it permissible to write the story in this way?'. The ainsi ('like this') in question is a matter not just of speculation, or of 'bearing false witness', but of narratological omniscience (and, by extension, everything that might be called 'internal' narrative). The answer is implicit, but complex: 'If it's a (fictional) story, yes… if it's history, then no.' It is typical of Voltaire that he should employ a pointed ambiguity to force an ironic wedge of definition between polysemes (he does just this with the eponymous word honnêteté throughout the piece, and the same effect is produced with the play on personnage in the epigraph above). This is how he means to break up Histoire and histoire.

The Impossible Letter 19

It is in response to this attack that La Beaumelle appears to have written the last of his apocryphal letters of M me de Maintenon (reproduced below). In response, that is, not merely to Voltaire's attack on him in particular, but much more importantly to Voltaire's violent separation of history and storytelling. For La Beaumelle, the pretext for this separation -the existence of an ideally objective historical perspective, which prescribes an analogous narratological point de vueappears fantastic. His is an audacious act of defiance in the circumstances: a détournement which is not content to ignore Voltaire's stylistic and methodological dictates, but which insists upon calling into question both the historical reality of the character in question and the very possibility of a non-fictional account of her life. Rather than defend the authenticity of his version of M me de Maintenon, La Beaumelle champions his fictitious character against the factitious accounts that were circulating at the time. He does so on the grounds that fiction is (as it were) her reality, and by allowing her to say this for herself. In doing so, La Beaumelle avoids restoring the Aristotelean paradigm in which 'poetry' is superior to 'history' (as Racine may have been tempted, eventually, to do), prefering a ludic(rous) pastiche which favours the Baroque mercuriality of style found in Paul Scarron's Roman Comique over the classical 'unities' (of time, place and action) in the theatre of Racine and Corneille. 20 His achievement here -an extraodinary act of intellectual generosity when one considers the habitual villification of M me de Maintenon amongst his fellow Huguenots -is to represent this defiance of History and its classical 'perspective' as emergent from the writing of M me de Maintenon herself. If this is the conclusion La Beaumelle has come to in his reading (and his writing) of her (purloined) letters, it is a justifiable and pleasingly subtle one. However unjustifiable and unsubtle the letter itself might appear as a forgery (as a Froissartean oeuvre of historiopoesis), we can at least borrow this conclusion from between its lines:

At the moment of its inception (if we can allow a historical perspective to converge on such an arbitrary point de fuite), the divorce of history and fiction in Enlightenment ideology is forced -by the demurral of the libidinously objectified protagonist whose intimate reality it pretended to uncover -to fold back in upon itself like a letter that should never have been opened in the first place… M me de Maintenon au Roi. Février 1689. Saint-Cyr 21

Sire, 22

Tout ce que j'étois avant que votre Majesté m'ait dépliée est écrit sur cette feuille. Votre Majesté a rompu le vermillon de mon cachet le plus confidentiel. Vous tenez, Sire, dans vos mains, mon entièreté; vos doigts touchent les plis 23 de mon être tremblant. Or, tremblante, je me suis résolue à être épanouie par votre Majesté;: à vous laisser voir, à vous laisser lire, ce qui n'aurait jamais du être inscrit à l'encre, ni gardé à l'esprit. 24 Sa Majesté se rendra compte que l'écriture n'est pas la mienne, 25 sans même compter du style. Il est le gribouillage estropié de mon feu mari. La lettre fut écrite pendant ce drame affreux qu'il savoit obscurément seroit sa maladie ultime. (Or j'ignore comment un tel état puisse être distingué, sauf avec le recul, par une telle âme en peine. 26 ) Vous découvrirez, Sire, qu'elle fait planer le doute sur l'honnêteté, la réalité même, de l'insaisissable 27 personnage, veuf et vierge, avec lequel vous ne vous mariâtes, je crains, qu'afin de la connoître. 28 Quoique je sois depuis venue à connoître beaucoup plus intimement la déception qu'elle dévoile : mieux que la lettre puisse traduire, ou que son auteur pût imaginer; vous devez croire, Sire, que je savois quasiment rien de son contenu jusqu'à ce que je l'aie ouverte il y a une petite quinzaine. D'ailleurs, elle contient quelques faits que j'étois vraiment consternée d'apprendre. Votre Majesté ne pourroit guère être aussi boulversé lorsque vous lirez l'histoire de ma vraie parenté avec M. Scarron.

Il se pourrait bien que votre Majesté demande comment j'eusse pu garder une telle lettre aussi longtemps sans l'ouvrir. La réponse est simple. Je l'ai oubliée. J'ai le don de l'oubli, 29 il s'avère, et j'avoue que j'ai perfectionné ses techniques pendant de longues années d'une vie de pure comédie. 30 L'oubli est le don le plus précieux de la comédienne. La réussite comique ne repose point sur la capacité d'apprendre par coeur les vers d'un poète et les moeurs d'un personnage, mais plutôt sur l'oubli : l'oubli de tout autre vers et phrase, le moindre mot qui ne convient point son rôle, l'oubli surtout de ses antécédents éventuels. 31 Au contraire, le coeur de la comédienne parfaite est vide comme le tube de M. Pascal avant que le mercure de la fiction monte et montre la hauteur de sa passion prétendue, la mesure de sa feinte modestie. 32 Coeur vide, en plein oubli de moi, je deviens ce à quoi les autres veulent croire, ce qu'ils aimeroient rêver que je sois. C'est ainsi que je me suis attirée le soutien de Mme de Montespan, l'admiration de la cour, les bonnes grâces de votre Majesté, et autant d'ennuis mélancoliques. Par force j'eus oublié la lettre. J'eus quasiment oublié l'histoire qu'elle raconte : à savoir, mon histoire à moi. Mais je n'eus point pu oublier le péché de l'oubli lui-même. Le vide dans mon coeur devenoit absolu. J'étois vide : non juste creuse comme l'esprit-dejatte 33 qui écrivit cette lettre, même point vidée de mon air comme le vide du philosophe, mais remplie plutôt à ras bord d'un vide complet, 34 un vide sans portée ni dimensions. Voici le vide qu'abhorre la nature. Voici le grand péché d'omission de l'hypocrite. Voici, Sire, ce que je souffrois avant que notre amitié intime ne se soit ébauchée.

Dieu m'accorda depuis la capacité d'être creuse. Votre Majesté remettra en cause peut-être la valeur d'une telle découverte. Il comprendroit, cependant, s'il savoit à quel point cela me réjouit le coeur. Pour moi, cette capacité est plus que précieuse. Mon coeur n'est plus vide comme il l'étoit avant; il n'est plus une enceinte 35 du néant. Je sens le creux au centre de mon être et je sais que je peux enfin briguer un coeur rempli. Je l'avois brigué, au début, mais je ne connoissois que la première rosée tendre de son aube qui l'imprégna en votre présence, Sire, sous votre regard. Et le souvenir des dimensions de mon intérieur, distinguées de cette manière, me laissa un creux dans le coeur en votre absence : un creux de bonheur poignant que j'eus presque renié. Dedans ce creux je trouvoi l'amour. Ce creux définit mon estime pour votre Majesté : j'ai trouvé de l'espace où il n'y avoit rien.

Dès lors je me rappelai. Je ne savois que trop bien ce que je cachois, et ma mémoire s'attarda, fatalement, sur cette lettre épouvantable adressée à votre Majesté. Elle me dans mon écritoire. Si leger fût son bout de papier, si mince son sceau, elle m'appesantissoit. Enfin, suite aux méprises de l'interprétation de l'Esther de M. Racine, 36 entourée de ces pages vierges : les murs et filles de Saint-Cyr, je l'ouvris.

Qu'est ce que j'avois imaginé? Quelque chose de véritablement terrible : l'exultation du formateur de l'assassin de votre Majesté, peut-être ; la recette du poison que je lui eus administré? A sa place je ne trouvai qu'une insulte monstrueuse en prose acerbe 37 , un gest grossier et dérisoire : aussi grotesque que sa Mazarinade. 38 Je n'avois point besoin, bien sûr, d'être rappelée que l'histoire fantastique de mes antécédents avoit été une fabrication de Scarron (la naissance en prison, le père faussaire athée, l'éducation hérétique, l'exil aux Antilles, la rééducation par les Soeurs, le mariage inconsommable avec le poète cul-de-jatte, mes amitiés avec votre reine et votre maitresse, le rôle de l'Esther de la France) : un conte absurde. Ceci n'étoit que trop clair pour moi. Je connoissois bien la chanson. Néanmoins, j'eus oublié jusqu'à quel point cette absurdité eut été déliberée de la part de mon soi-disant mari, comment son ironie fut à l'origine de cet invraisemblance, avec quelle prévoyance insolite il eut comploté mon rapprochement à votre Majesté. La phrase, pour moi, la plus frappante étoit : « Maintenant, mon vieux, Françoise aura gâté tous tes vains désir, et tous tes bâtards vains, par son refus de tout et de rien : et de rien, c'est mon plaisir. » Ici retrouvai-je tout ce que M. Scarron fut devenu par la fin de sa vie : son impudence habituelle, son alchimie anormale de la langue française, sa parodie sardonique des honnêtetés courtoise, et ses prédictions lancinantes de l'avenir.

En suite, son 'dernier testament'. L'exigence que votre Majesté soit témoin d'une telle liste de legs, sans queue ni tête, l'écoeurera 39 sans doute. Par quel processus puisse-t-on hériter ces songes fantasques, ces ambitions démesurées, ces connoissances inconcevables, cette nommée « puissance universelle de ma comédie,»? Et puis, « Le reste je donne au Roy. Bref, je luy donne le tout : ma fronde 40 mignonne, ma belle Esther elle-même. » Enfin, par quelle autorité ose-t-il faire un don de moi? 41 Par quelle méchanceté avoit-il me caché mon lignage? Vous ne devrez point imaginer, Sire, la licence que ce recel de notre parenté lui eût fourni de me regarder avec une convoitise non déguisée! 42 Or, je ne peux pas me présenter auprès de votre Majesté comme la seule victime de cette affaire. C'étoit la vision de Scarron d'une intervention impensable dans l'histoire de l'avenir qui forgea le drame, 43 mais je me suis façonnée en son vaisseau, 44 en son executante. Si la farce scandaleuse prévue par le feu satiriste a été évitée, c'est grâce à Dieu, Sire, et à vous. Malgré mon péché grave, mon ambition adoptée de conquérir votre cour, vous avez conquis par la suite mon coeur. Maintenant, j'ose vous envoyer cette lettre. Maintenant j'ose avouer le tout, et je vous jure de mettre mes capacités au service du plus grand bien dorénavant. Je consacrerai mes talents au souverain et à Saint-Cyr. Je me mettrai entièrement à votre service, Sire, et celui de mes filles, et à la merci éternelle de Dieu et de mon Roi. Je reconnais que je mérite être traitée comme le plus vil et le dernier de tous vos sujets. Pardonnez-moi donc, Sire. Faites de moi ce qu'il vous plaira. 45 Renonçant à tout droit de ne jamais me reprendre, je m'abondonne, Sire, pour toujours, à votre bon plaisir. ) This is as good an example as any of how valuable a piece of daft gossipy historical fiction can be to an appreciation of the culture it describes. The book is sheer fantasy, no doubt, but the portrayal of a woman struggling unconvincingly through the consequences of her own grotesque, self-conscious unreality could not be more pertinent. 'Bad writing', 'tawdry bandwagoning', fair enough: but the criticism that 'this is not the real Diana' is risibly deluded.

momentum is a runaway train of an etymon. In Classical Latin, the most common expression equivalent to the Anglo-French adjective 'important' was magni momenti: 'of great moment'. That this might just as easily be a reference to a crucial instant, a powerful force or a considerable weight is indicative of the complexity of the metaphysical metaphor. The ambiguity is extended by the logical connection between the salient and the memorable. And this is fuelled by a slip of the ear. The imperative of the verb to remember in Latin is memento. It is via a mistaken nominal use that we come, in English, to speak of a 'memento', and especially of a memento mori. Or did you say momentum mori? Grammar quails at the death. Voltaire, in seeking to upset the momentum of La Beaumelle's memento of Madame de Maintenon, wants to divide the act of (fictional) remembering from the (impossiblly objective) historical moment. You cannot speak as if you remember because you were not there at that point in time to make that memory... Might we fall into the same trap by retracing the Englightenment's hierarchy of historical moment to this moment of inception?

9

See for example, the fictional autobiography of M me de Maintenon: l'Allée du Roi, by Bluche's friend Françoise Chandernagor (Paris, Editions Julliard. 1981) 10 'This Secret Diary of Louis XIV presents us with all the de facto characteristics of authenticity: verified events, exact dates, plausible political and psychological analyses. All of the important facts of a great reign are recounted or evoked here: from the disgrace of de Fouquet to the revocation of the Edict of Nantes, from the mystery of the Iron Mask to the secrets of the alcove… all of the history of the Grand Siècle as seen by the person who lent it its stature: the Sun-King in person.' 11 'Madame de Maintenon seems to me to be exemplary not least because she poses the question of the gift of time -and the rest -from the perspective of a woman and a grande-dame. She who played the role of Louis XIV's "sultan(a) of conscience" was at the same time (the configuration is rarely fortuitous) an outlaw and the very figure of the law. Before becoming, upon the death of the Queen, the morganatic wife of the King (thereby excluded from all noble titles and rights -and the word "morganatic" says something of the gift, the gift of the origin: it comes from the low Latin morganegiba, gift of the morning), she had been bringing the Sun King back to his duties as a husband (by distancing him from Madame de Montespan, whose protégée she had been) and as a Catholic king (by restoring austerity to the court, by encouraging the persecution of the Protestants -even though she herself was raised a Calvinist -and by lending her support to the revocation of the Edict of Nantes). She who took so much trouble over what one should give and take, over the law, over the name of the King, over legitimacy in general, was also the governess of the royal bastards, a promotion she no doubt owed to the protection of Madame de Montespan. Let us stop where we should have begun: when she was just a child, she experienced exile in Martinique and her father, Constant, was arrested as a counterfeiter. Everything in this life seems to be struck with the most austere, the most rigorous, the most authentic die of counterfeit money.' 12 'History is a sacred thing. What's that! you transcribe the letters of a historical figure, of a great King, of a hero, and you mix in your own turns of phrase and your own thoughts, without telling me! I believe I'm studying Frederick, I apply myself to criticizing or commending him, I rely upon his word and his authority, and I'm duped, I've been fooled, I have nothing in my hands except a 'La Beaumelle': counterfeit money stamped with the King's effigy! ... we will continue, more than ever, to say a 'La Beaumelle' to refer to the faithless editor par excellence.'

13 Any such conclusion is clearly open to the critique of being premised upon an impossible attempt to transcend hindsight itself by reading the 'originals' in an unimaginable historical space evacuated of all knowledge of the fakes. The Impossible Letter 19 The letter is published here for the first time. The original manuscript, which was included in a collection of La Beaumelle's German correspondence and other miscellanea, was destroyed, along with the rest of the archives of the Frankfurt am Mein city library, by an allied bombing raid in 1944. The only copy (I know of) was made by a student at the Goethe University of Frankfurt in 1934 who was researching a dissertation on the French literary influence on the court of Frederick II of Prussia, which he later abandonned for the politically less controversial study of medicine. The copy was made in one of the volumes of his student journal which -being a fastidious kind of person -he managed to preserve throughout the war and even to transport out of Germany (with the rest of his diaries) in 1945. I am rather ashamed to admit that these diaries were inherited by an ex-partner (his grandchild) on his death, and subsequently became subject to a rather squallid wrangle over debts and ownership of books when the relationship broke up. The diaries have since been restored to their rightful owner. The decision to publish without naming the student in question has been taken jointly.

There are obviously doubts as to the letter's authorship. The French, whilst sophisticated, is sometimes idiomatically bizarre and shows signs of being (at the very least) manipulated or supplemented by a nonnative (if relatively expert) writer of the language. It is conceivable that the student made the whole thing up. It seems more likely, however, that the idiomatic oddities derive from the mistranscription of a text that was poorly preserved or difficult to decipher for a young scholar with little experience of dealing with historical manuscripts. In any event, in keeping with the basic pro-fictional tenets of this paper, I have decided to go along with the story that this is the unpublished work of La Beaumelle. The only other influence my expartner's grandfather has had on this study is the suggestion that the letter is a response to Voltaire's Honnêtetés. The theory that it represents a rejection of Voltaire's quasi-definitive separation of Histoire and histoire is my own. 20

It is partly, perhaps even principally, in protest at their flagrant flouting of these unités that Voltaire denounces Shakespeare's tragedies and histories as 'barbaric'. (Lettres philosophiques...) Your Majesty will realise the hand is not my own, much less the style. It is the hobbled scrawl of my late husband. The letter was written during that awful drama which he somehow knew would be his final illness. (Though I do not understand how such a state can be distinguished, except in hindsight, by such an afflicted soul.) You will discover, Sire, that it casts into doubt the honesty, the very reality of the elusive character, widowed and untouched, whom I fear you married just to know her. Even though I have come to know much more intimately the deception it unveils, better than either the letter can convey or its late author could imagine, you must believe, Sire, that I knew almost nothing of its contents until I opened it less than a fortnight ago. Indeed there are facts that I was truly appalled to learn. Your Majesty can scarcely be as distressed when you read the story of my true relationship to Monsieur Scarron.

M
Your Majesty may well ask how I could have kept such a letter for so long without opening it. The answer is simple. I forgot it. I have a gift for forgetting, it turns out, and I confess that I have perfected its techniques during the long years of a life of pure theatre. Forgetting is the most precious gift of the actress. Theatrical success does not rest upon the capacity to learn by heart the lines of a poet or the aspects of a character, but rather on forgetting: forgetting every other verse or phrase, the slightest word that is unsuited to her role, forgetting above all the possible events of her past life. On the contrary, the heart of the perfect actress is as empty as one of M. Pascale's tubes before the quicksilver of the fiction rises and represents the height of her pretended passion, the measure of her false modesty.

With an empty heart, and completely forgetting myself, I become that which other people want to believe in, that which they would like to dream that I am. It is like this that I've garnered the support of Mme. de Montespan, the admiration of the court, the attentions of Your Majesty, and so many melancholic worries. Perforce, I had forgotten the letter. I had virtually forgotten the story that it tells, namely my own history. But I had not been able to forget this sin of forgetting itself. The emptiness in my heart was becoming absolute. I was empty: not just hollow like the lame-brain that wrote this letter, not even emptied of my air like the philosopher's vacuum, but rather full to the brim with a complete void, a void with neither scope nor dimensions. This is the void that nature abhors. This is the great sin of omission of the hypocrite. This, Sire, is what I suffered before our intimate friendship began to find a form.

Since then, God has granted me the capacity to be hollow. Your Majesty will perhaps question the value of such a discovery. He would understand, however, if he knew to what degree it has gladdened my heart. For me, this capacity is more than crucial. My heart is no longer empty like it was before; it is no longer a well of nothingness. I feel the hollow at the centre of my being and I know, at last, that I can aspire to a full heart. I had aspired to it, at the beginning, but I knew only the first tender dew of its dawn that impregnated it in your presence, Sire, beneath your gaze. And the memory of my interior dimensions, marked out in this way, left me with a hollowness in my heart in your absence: a hollowness of poignant happiness that I had all but renounced. In this hollowness I found love. This hollowness defines my estime for Your Majesty: I have found space where there was nothing.

From that point on, I remembered (myself). I knew all too well what I was hiding, and my memory lingered, fatally, upon this dreadful letter. It haunted me in my escritoire. However flimsy (was) its slip of paper, however mean its seal, it weighed me down. Finally, after the disturbances of the performance of M. Racine's Esther, surrounded by those virgin pages, the walls and girls of Saint-Cyr, I opened it.

What had I been imagining? Something truly terrible: the exultation of the trainer of the King's assassin, perhaps; the recipe for the poison I had administered to him? Instead I found only a monstrous insult in acerbic prose, a rude and risible gesture: as grotesque as his Mazarinade. I had not needed, obviously, any reminding that the fanciful story of my past had been a fabrication of Scarron's (the birth in prison, the atheist counterfeiter of a father, the heretical education, the exile in the West Indies, the rehabilitation by the nuns, the marriage that could not be consummated with the poet-cripple, my friendships with your queen and your mistress, the role of France's Esther): an absurd fairytale. It was all too obvious to me. I knew the song well. Nevertheless, I had forgotten just how much this absurdity had been deliberate on my so-called husband's part, how his irony had been the origin of this fantasy, with what uncanny foresight he had plotted my wooing of Your Majesty. The phrase which for me was the most striking was : "By now, my old friend, Françoise will have spoiled all your vain desires and all your vain bastards with her refusal of everything and of nothing: and it's nothing, it's my pleasure." Here I found everything that M. Scarron had become by the end of his life: his habitual insolence, his unnatural alchemy of the French language, his sardonic parody of courtly manners, and his piercing prediction of the future.

And then, his 'last will and testament'. The demand that Your Majesty be witness to such a ludicrous list of bequests (that one can make neither head nor tail of) will no doubt sicken his heart. By what process might one inherit these fanciful dreams, these immoderate ambitions, these inconceivable types of knowledge, this "universal power of my comedy" he mentions? And then "The rest I give to the King. In short, I give him everything : my lovely little fronde, my beautiful Esther herself." By what authority does he dare to make a gift of me? By what kind of spitefulness had he hidden my lineage from me? You should not try to imagine, Sire, the license that this concealment of our kinship afforded him to look at me with undisguised covetousness.

And yet, I cannot present myself to Your Majesty as merely the victim in this affair. It was Scarron's vision of an unthinkable intervention in the story of the future that forged this drama, but I fashioned myself into its vessel, its performer. If the scandalous farce envisaged by the late satirist has been avoided, it is thanks to God, Sire, and to you. Despite my grave sin, my adopted ambition to conquer the court, you have in turn conquered my heart. Only now do I dare send you this letter. Only now do I dare confess to everything, and I vow to you that I will henceforth use my skills to the benefit of the greater good. I will devote my talents to the sovereign and to Saint-Cyr. I will place myself entirely at your service, Sire, and that of my girls, and at the eternal mercy of God and of my King. I realise that I deserve to be treated as the vilest and basest of your subjects. Therefore, forgive me, Sire. Do with me what you please.

Renouncing all rights to ever regain my liberty, I abandon myself forever to Your Majesty's pleasure. 22 Sire : the standard manner of addressing the King is used in this letter as a kind of glue to hold together otherwise unruly syntactical constructions. Lionel Trilling, in Sincerity and Authenticity (London OUP 1971 p.12), a work that obviously adheres in patches to this kind of pastiche (but also turns up sticking points), tells the charming story of the false etymology for the word sincère that was still something of a neologism in the 17 th century, at least as a word used to refer to a human quality. He claims that it was thought to have derived from the Latin sin cera 'without wax', the idea being that it meant something like 'coherent', as in : 'not a botched job stuck together with wax'. This word, in this context, is difficult to unstick from the nearly homophonous 'Saint-Cyr' (a problematic point de repère of pedagogic sincerity if ever there was one : participative jeux de rôle and didactic comédies having been an integral part of the school's curriculum). The fact that the King's honorific title is a pun on the stuff that holds together a definitively untrustworthy collage or pastiche, and also that which seals the letter (always a potentially treachorous medium in the 18 th century fictions and histories from which La Beaumelle's oeuvre cannot be detached), would seem to suggest a reading which ironizes the King's role as the constitutional gel that holds together the court and the kingdom. 23 plis : 'folds'. "Mme de Maintenon dit et écrit en perfection. Tout tombe juste, il n'y a pas un pli dans ce stylelà" (Sainte-Beuve, Causeries du lundi, t.4, 1852, p.387). Despite Saint-Beuve's sycophantic eulogy to her agelessly ironed prose, Maintenon's life and letters have been so often folded, unfolded and refolded that it's hard to see beyond the pleats. Deleuze's work on folds in the Baroque aesthetic crosses (or creases) the mind, and La Beaumelle's performance here is certainly Baroque in its 'architecture of vision' : its unfolding and refolding story of history. In fact, the sinfully sorry state in which Maintenon claims to have languished -one of a shell without internal dimensions -is almost the total opposite of the Leibnizian notion of the monad which Deleuze so cogently demonstrates is analogous to Baroque architecture : 'La monade est l'outonomie de l'intérieur, un intérieur sans extérieur.' (Le pli : Leibniz et le Baroque. Paris, Les éditions de minuit 1988 p.39)

The tranformation she undergoes into a positively inflected âme creuse (a 'hollow soul') by submitting herself to (en se pliant à) 'the gaze' of the King, is therefore a kind of inversion of the effect of the trompe-l'oeil of the Baroque interior : 'une pièce en marbre noir, où la lumière ne pénètre que par des orifices si bien coudés qu'ils ne laissent rien voir du dehors' (1988 p.39) The ramifications for a vision of history, and for the historical perspective, are manifold. La Beaumelle is problematizing (by multiplication) the implicitly classical historical perspective suggested by Voltaire's proscription of the 'interior' point of view of prose-fiction (both the interior of the bed-chamber and the interior of the historical character's mind or heart). When I say that a Baroque view of history problematizes the classical historical perspective, I mean to take this as far towards its literal vanishing point as it will go. The classical historical perspective is very precisely analogous (in the temporal dimension) to that convergent hierarchy of organisation (in the spatial dimension) that Foucault shows is multiplied to infinte regression by Valazquez's Las Meninas. Epanouie, in this context, is openly ironic.

As for the style, unusually for La Beaumelle, whose fine mimicry of her prose is usually grudgingly acknowledged (though not by Voltaire), the style of this letter is unconvincing as a pastiche of Mme de Maintenon. It is far too sentimental and showy for such a clear and level-headed writer. Syntactic oddities, tensual pedantry and pointedly outdated vocabulary suggest a writer trying too consciously to adapt his style. Might this be premeditated -sans même compter du style : revelation rather than disguise? 24 This paragraph follows a seemingly dialectical pattern of thesis, antithesis and synthesis with regard to the pronominal address of the King : a pattern which is regularly repeated in the letter. The first two sentences indirectly address 'votre Majesté' in the third person. The third switches to the polite second person form : 'Vous tenez, Sire'. And the final sentence mixes 'Votre Majesté' with the 'vous' form. There is no chance of La Beaumelle being influenced by Hegelian 3-part dialectic. This observation is necessarily anachronistic. It is nevertheless suggestive to a modern reader of a proto-dialectical approach to the synthesis of history and fiction (as poesis). This reinforces the general point that the identity of the King is revealed to be as implicitly 'synthetic' as that of his 'ungraspable' morganatic wife (see note 27 insaisissable). 25 We are to imagine an enclosed letter written by Paul Scarron addressed to the King. It is this second letter, and not the one we are reading, to which the first paragraph refers as 'my entirety'. It seems unlikely La Beaumelle ever intended to write this letter, prefering to leave readers to imagine its contents : especially the answer to the riddle of how exactly Françoise is related to the poet. In turn, we are left in no doubt as to the unattainability of such a juicy historical tit-bit. This is particularly obvious given the fact that the instigatory act described -of the King rupturing the seal on a long-unopened letter, and thereby gaining access to a hidden secret -reveals itself to be an impossible anachronism when Mme de Maintenon claims to have read the letter a fortnight previously. Further doubt is cast, therefore, upon the integrity of the Sire/cire. 26 âme en peine : Paul Scarron was severely disabled. He suffered near paralysis and chronic pain as a result of what was probably acute rheumatoid arthritis throughout his marriage with Françoise. La Beaumelle claimed this to have been the result of Scarron submerging himself in the icy river Huisne in Le Mans during the carnival of 1638 whilst hiding from a mob who had been incensed by his indecent appearance (he had supposedly covered himself with nothing but beeswax and feathers). It is unclear if La Beaumelle was trying anything more than to dress up Scarron's history with a colourful (if cloying) fiction. Given Scarron's medical history, it is perfectly reasonable for the young woman to be surprised that he was capable of recognizing that the illness in question would be his last. The phrase La Beaumelle has her use, however -âme en peine -has damning overtones. It means 'punished' or 'lost' soul, and was the phrase used in French to refer to the shades beside the Styx in Dante's vision of hell. She is implicitly suggesting that Scarron's predictive ability is demonic (more specifically, necromantic). As such this can be seen as an ironic confession of La Beaumelle's supernatural ability to see the 'future' from a historical perspective. 27 insaisissable : just as 'grasp' does in English, saisir in French suggests both the literal act of taking into the hand and the metaphorical act of knowing or understanding. The role that La Beaumelle's Maintenon ascribes to herself, as the 'virgin widow', is therefore one in which she is both the unattainable (literally untouchable) object of sexual desire and the unknowable mystery at the centre of an obscure history. A crucial further level of irony is revealed by the pun on maintenant. The Horatian cliché carpe diem notwithstanding, the 'seizable moment' is as false a notion as Derrida's ridiculed 'gift of time'. Any attempt to hold on to time inevitably sees it slipping through our fingers. The same, for La Beaumelle, is true of this 'ungraspable [historical] character.' connoître : The play on knowledge and sexual conquest continues. French has precisely the same 'biblical' euphemistic sense of the verb 'to know (a person)' as English. Maintenon is teasing the King by suggesting that he only married her so that he could have sex with her: the least attainable woman in court. What he ends up with is an 'unknowable' woman: a hollow victory in more ways than one. 29 le don de l'oubli : It is difficult to capture the key word here -the noun oubli -in English. There is no very stable nominal form of 'to forget', and the French word is weightier and more polysemous than the concept of forgetting. It recalls ideas of omission and neglect and is cognate with 'oblivion'. To say one has le don de l'oubli is almost to lose oneself in an oxymoron. The idea is clearly related to that of the vide ('void' or 'vacuum') she explores in the next paragraph. 30 vie de pure comédie : she does seem to be singling herself out here, as particularly histrionic, but the idea of society being composed of a series of roles played by social 'actors' -especially of the awareness of this as being a socially liberating bourgeois ideology -is a theme of capital importance for La Beaumelle's contemporaries. Lionel Trilling chooses Diderot's Rameau's Nephew as representative of a watershed in this ideological decline of noble sincerity : 'it is agreed between Diderot and the Nephew, between the Moi and the Lui of the dialogue, that everyone in society, without exception, acts a part, takes a 'position', does his dance, even the King himself, 'who takes a position before his mistress and God: he dances his pantomime steps.'' (1971 p.31) La Beaumelle's Maintenon implies precisely this complementarity of histrionic social (and sexual) relations between King and mistress with her statement of the opposite. 31 ses antécédents éventuels : this is a strange, complex and ambiguous phrase with powerful resonances in a debate about history, especially in its strongly emphasised position as 'that which the actress must forget, above all'. It could either mean 'the contingent past events of her life' or else 'one's possible ancestors.' 32 The debate between Descartes and Pascal concerning the possibility of the vacuum is a crucial philosophical and scientific discourse in mid 17 th century France. It pits empiricism against rationalism and plays a role in the bloody religious controversies of the period: Aristotle's assertion that la nature abhorre le vide having been adopted almost as dogma, to the extent that believing in the vacuum was considered sinful. The impact upon the tumultuous regency zeitgeist of Pascal's Expériences nouvelles touchant le vide 1647 (with its provocatively ambiguous title : 'New experiments concerning the vacuum' or 'New experiences of touching the void') cannot be underestimated. Despite her usual avoidance of the physical sciences, it is not beyond the realms of possibility that Maintenon could have had an in-depth knowledge of the debate from (for example) a theological perspective. What is less likely is the sophisticated (though philsophically shaky) means via which La Beaumelle has her implicitly transcend the debate in the ensuing exploration of le vide. He has her differentiate three distinct types of vide : hollowness (that which must be filled), vacuum (that which has been totally evacuated) and void (that which contains a total absence, even of capacity). At first she likens the emotional emptiness of her heart to Pascal's vacuum, but she goes on to correct herself and to confess to a genuine spiritual void ('the real emptiness that nature abhors') in her heart. Her rediscovery of love, and of herself, is defined in turn as a recategorization of the emptiness of her heart: from void to hollowness. It is perhaps the key point of my thesis that La Beaumelle is implicitly using Maintenon as an index of history itself, and is therefore suggesting that his fictitious Maintenon/history (as a hollow vessel) is an improvement on the factitious Maintenon/history that has been rendered void by Voltaire's attempt to divorce it from the imaginative processes that give it 'interior dimensions.' 33 esprit-de-jatte : The term used in the seventh paragraph below, cul-de-jatte, is an insulting contemporary term for an amputee or somebody who can't walk. Literally, it means 'bowl-arse'. The neologism here means something like 'bowl-brain' and suggests, unfairly for such a witty target, intellectual vacuity. Esprit is polysemous, though, and it is obviously spiritual hollowness, rather than air-headedness, that she accuses Scarron of. It is perhaps important that the string of references to hollow vessels, in the context of an insistence upon infinite but unattainable promise, carries connotations of the Saint-Graal (the 'holy grail') in La Beaumelle's letter. 34 remplie plutôt à ras bord d'un vide complet : the paradoxical nature of this sentence is rather overplayed by La Beaumelle. Even the word complet, ostensibly employed as a synonym of total, connotes 'fullness'. To say that you are 'filled to the brim with a replete emptiness' is to be much more histrionic and verbally playful than Maintenon ever reveals herself to be in what we suppose to be her unfabricated letters. The absurdity of this, however, might be La Beaumelle thumbing his nose at Voltaire's demand that he expurgate from his history all speculative narration of Maintenon's inner thoughts. 35 enceinte : 'enclosure'. The phrase enceinte du néant, is basically synonymous with what she has previously called le vide absolu. Enceinte, however, is formed from the adjective meaning 'pregnant' and the phrase might quite easily be glossed as 'a barren womb'. La Beaumelle is not letting the reader forget his quasi-sexual desire for Maintenon (there can be no doubt that the reader is implicitly male, nor that the text is rather misogynist).

She is to be a potential 'mother' for the historian's libidinous ambition to engender his own history of Louis XIV: a role at odds with her childlessness and her infamous (and infamously doubtful) virginity. The prurient undertones abound, of course. A contemporary reader would naturally decode coeur and creux as vagin. 'The first tender dews that impregnated it in your presence, Sire' hardly requires a gloss. 36 The likening of Maintenon to Esther, first suggested by Racine's didactic moral play written in 1688-9 (at Maintenon's behest) for performance by the older girls of Saint-Cyr (girls of marriagable age, it should be noted, who made their débuts in this dubious fashion), is quite commonplace in the 18 th century. Ferriol D'Argental, for example, writing to Voltaire on the subject of La Beaumelle's alleged theft of the Maintenon correspondence from Louis Racine (18 th Dec. 1782), says of her « De l'ignorance, de la faiblesse, de la fausseté, de l'ambition, du manége, des messes, des sermons, des galanteries, des cabales, voilà ce qui compose une Esther. » (Lavalée op. cit. p. xvii). It is not simple, however. There is obviously an implicit critique of Maintenon, by comparison with Esther, but it is not clear whether this is because she is too much like Esther (the young second-wife of the King from a disadvantaged background of religious marginalisation has inveigled her way to a position of supreme influence) or because she doesn't measure up to Esther (in that she fails to save her 'people' from the Dragonnades). D'Argental clearly means the former, though there is a witty undertow of 'the fakery of her composition' in his sentence, but Racine's covert Jansenism suggests he might have thought the latter. It is crucial that the Book of Esther is not merely a story about religious persecution, but also a focus of the contemporary religious controversy in Western Europe. Martin Luther's objection to the Megillot scrolls of the Old Testament was well understood, and his singling out of Esther ('which despite their inclusion of it in the canon deserves more than all the rest in my judgment to be regarded as noncanonical' [letter to Erasmus on Free Will]) was subject to heated protestant polemic. Consequently, one of the major differences between Protestant and Catholic bibles of the period was the heavy expurgation in Calvinist versions of the apocryphal Greek content (which had inserted theological glosses and references to God into the Catholic text). La Beaumelle, as a Protestant, would be well aware of this controversy. He would also be more sensitive than most to the analogies that had been drawn by Racine. The added dimension of the interpolation of an apocryphal text is also likely to have appealed to the writer whom Voltaire had pilloried as the 'éditeur infidèle par excellence', as Sainte-Beuve will have it (see note 12 and French original). His great coup-de-théâtre in this letter is to cast Paul Scarron as a kind of Machievellian Mordechai and thereby to steal Jean Racine's thunder... just as he was supposed by D'Argental and Voltaire to have stolen Louis Racine's collection of Mme de Maintenon's letters.

37

insulte monstrueuse en prose acerbe : This paragraph provides good internal evidence that this letter is La Beaumelle's rejoinder to Voltaire's attack upon his historical credentials in les Honnêtetés Littéraires. The gambit of the rhetorical question, 'What had I been imagining?', is a direct invitation to the reader to indulge in the same imaginative recreation of a historical character's inner life that prompts Voltaire to the gaping rhetorical question at the heart of his appended 'Lettre à l'auteur...' : Est-il permis d'écrire ainsi l'histoire? What ensues can easily be read as a thinly veiled critique of Voltaire's Honnêtetés : so thinly veiled, indeed, that it refers to the 'sardonic parody of courtly honnêtetés ['manners' or 'etiquette']'. Phrases like prose acerbe and impudence habituel could easily be applied to Voltaire's work, and the term conte absurde (absurd fairy-tale) is lifted directly from Voltaire's conclusion to his 16th Honnêteté : "La Beaumelle. C'est lui qui a falsifié les Lettres de Madame de Maintenon, et qui a rempli les Mémoires de Maintenon de contes absurdes et des anecdotes les plus fausses."

[1767] La Beaumelle wittily finesses this idea of fairy-tale absurdity by attributing it, firstly, to what we might want to call the 'factitious' history (the conventional wisdom) of Maintenon's life, and secondly to the conscious irony of a writer whose goal is satire rather than verisimilitude. Finally, the quotation from Scarron's putative letter is clearly a parody of Voltaire. It has his famous irony, his over-familiar use of tu with all and sundry and his prediliction for the traductio figure of polysemous repetition (vain and de rien).

38 Mazarinade : the Mazarinades were satirical poems written at the time of the Fronde (see note 40 fronde below) ridiculing Cardinal Mazarin. Scarron's is the most famous and one of the most vehement. It explicitly casts Mazarin as a bougre -both a heretic and a pederast -who has 'sodomized the state'. It took him a good deal of time and effort to rehabilitate his reputation after the Fronde, and it is during this period that he is widely suspected of training his young (virgin) wife as bait to lure back the admiring gaze of the noblemen of the court: a beautiful curiosity with a remarkable education. 39 écoeurera : this is a very unusual usage. The direct equivalent of the verb in English would be 'to dishearten', but this captures nothing of the sense of disgust and disdain inherent in the French. Even in La Beaumelle's time, this word is considered a grand mot and, in the context of supposed 16 th century French, it seems to grate violently against the tone. It is not really until the mid 19 th century that it became acceptable. A translation that captures this stylistic blip might be : 'The demand that Your Majesty be witness to such a list of nonsensical bequests will no doubt fuck him off.' We can only guess that La Beaumelle chooses it as a metonymic link to the anatomical idiom sans queue ni tête (with neither head nor tail), and in order to press the (ironic) point about this as a letter designed to 'touch the heart'. 40 fronde : it is obviously telling that this Scarron should describe his 'Esther' as his 'slingshot'. He is putatively applying the cliché of the modest weapon used by David to slay Goliath that had given the name to the parliamentary and baronial rebellions against the regency of Cardinal Mazarin and Anne of Austria to which Scarron had lent his public support. Despite forcing Mazarin into temporary exile, the Frondes were ultimately a failed attempt to introduce a constitutional monarchy in France. The upshot was the absolutist divine-right monarchy of Louis XIV, which Scarron is supposed by this letter (improbably) to have foreseen, and even plotted to bring down. By inserting a beautiful, faux-naive young spy into the court to instigate a debilitating scandal, the letter supposes him to believe he can start a revolution. 41 un don de moi : this idea of Scarron (who actually left considerable debts to his widow) bequeathing Françoise a long list of abstract concepts, Rabelasian in its absurdity, and then bequeathing Françoise herself to the King, clearly has much more to do with La Beaumelle's own joke on the illusiory importance of her morganatic status (as the disenfranchized wife of the monarch) than anything Maintenon herself would have wanted to express. The whole point of a morganatic wedding is that the pièce rapportée, the incomer, has no access to the sovereign's heritage. Any separation would leave Maintenon, with nothing more than she brought into the marriage and any children she had by the King would have no more right to power or property than those of a Mme de Montespan. Of course, both divorce and descendents are an academic consideration in this case, but La Beaumelle's implicit critique of Voltaire's desire to divorce fiction and history (and divide their properties and territories) is not. Maintenon herself seen as a present, in both senses -a gift (of time) at her own wedding -is an idea carefully unwrapped by Derrida's Donner le temps... All he reveals, ultimately, is the never-ending emballage of texts that envelope time. Like time itself, like history, this baroque Maintenon is an interminable game of pass the parcel. It is worth noting that between 1643 and 1652 Scarron had been involved in a costly and ultimately inconclusive court case regarding the interpretation of his father's will. Think Jarndyce and Jarndyce. 42 recel de notre parenté... convoitise non-déguisée : recel means 'concealment' here, but it is also the name of the crime of receiving (and hiding) stolen goods. It is likely that La Beaumelle means this as an encoded reference to his alleged theft of the letters of Madame de Maintenon held by Louis Racine, the son of the poet Jean Racine, who presumably had them as a result of his functions as one of the King's official historiographers (the other was Boileau). In any event, it is a typical flip-flop of the ideas of secretion and revelation that she should refer to : 'the license that this concealment of our kinship afforded him to look at me with undisguised lust'. The question of exactly what this 'kinship' is supposed to be -the most likely thing would probably be uncle and niece, but even father and daughter cannot be ruled out -is obviously also concealed by La Beaumelle's letter itself. This being the case, La Beaumelle can be understood as potentially playing the same ambiguous (implicitly incestuous) role of libidinous 'forger' of Françoise, and ironically admonshing himself for it. As historical readers, we are to infer that we (like the most obviously implied reader, Voltaire) are doing the same. I can't help reading in Nabokov's Lolita, which is a case of recel both in the sense that it is suspected of being 'plagiarized' and that it concerns a plagium : the theft of a dependant or slave -a 'kidnapping'.

43

une intervention impensable dans l'histoire de l'avenir qui forgea le drame. You could hardly invent a more apt expression of the ironic self-rebuke for a comic overstepping of the arbitrary temporal boundaries of history. 44 je me suis façonnée en son vaisseau : ''I have formed myself into its vessel.' The specific wording of her confession of her role in this (hi)story returns us to the idea of the hollow container. The contrite context notwithstanding, the fact that the reflexive in French has an ambiguous relationship to the idea of free-will suggests a rather passive gloss : 'I have been shaped into its vessel.' It seems to allude to Jeremiah 18 : "And the vessel that he made of clay was marred in the hand of the potter: so he made it again another vessel, as seemed good to the potter to make it. Then the word of the LORD came to me, saying, O house of Israel, cannot I do with you as this potter?... Behold, as the clay is in the potter's hand, so are ye in mine hand." [King James Version: Jer. 18 : 3-6]. As such it reliteralizes the notion of maintenant, the moment in time (and implicitly the history) that can be held in the hand and have its form literally manipulated. It also serves to introduce the final (somewhat blasphemous) turn, in which the concluding salutations appear to be lifted directly from a penitential prayer in the Quietist prayerbook written by Jeanne Guyon : Le moyen court et très facile de l'oraison... 1682. For Sire, read Seigneur. The joke is a denunciation, by the ironic hyperbole of addressing an act of total spiritual supplication to the King, of the divine right of the monarch... a (waxy) foretaste, almost, of De Sade. 45 de moi ce qu'il vous plaira : a direct quotation of the prayer of Madame Guyon. La Beaumelle has cut the line : je me donne à vous sans réserve ('I give myself to you without reservation'); otherwise the prayer continues exactly as she wrote it. He has probably made this cut to avoid excessive repetition of ideas, but it would have made a pertinent final link to Derrida's work on the gift, and the rest. Perhaps, in the circumstances, its omission can be understood as even more of a gift to the critical theorist than its imagined inclusion. The ambiguity of the verb faire in this last sentence of the body text, and the deferring / deferential future tense, are probably informative of what is expected of historical readers : 'do with me what you will', she says, or rather, 'make of me whatever will please you.' -i.e. l'histoire or else tout une histoire.

  14

  (Louis XIV -Jean-Pierre Kalfon -to M me de Maintenon -Isabelle Huppert, before having his way with her in a carriage flanked by her entourage: Saint-Cyr, film 2000. Writ./dir. Patricia Mazuy, after the novel: la Maison d'Esther, Yves Dangerfield, Paris, Grasset 1991)

table ?

 ?

	Est-il
	permis d'écrire ainsi l'histoire? 18
	(Voltaire "Lettre à l'auteur des Honnêtetés littéraires, sur les Mémoires de Madame de Maintenon,
	publiés par La Beaumelle" 1767 Source: Voltaire Oeuvres complètes en ligne -
	www.voltaire-integral.com/Html/26/11_Lettre_a_lauteur.html)

  My heart is empty; my mind is constrained : I play the role of the principle 'personage' of France [the protagonist / the most important figure]; but this is just a fictional character. I'm living nothing but a borrowed life.' Perhaps this term should not be used here. It is the standard translation in English (and, indeed, in French: sphère publique) of the 'public' side of Jürgen Habermas's widely used dichotomy of Enlightenment culture Privatsphäre and Öffentlichkeit (Strukturwandel der Öffentlichkeit. Untersuchungen zu einer Kategorie der bürgerlichen Gesellschaft. (Habil.),Neuwied 1962 [Neuauflage: Frankfurt a. M. 1990]). In fact, it is a rather grave mistranslation. The geometrically enclosed connotations of (the etymologically French) Privatsphäre are thrown wide open by the boundless implications of (the etymologically Germanic) Öffentlichkeit. One is a sphere, the other certainly is not. c.f. notes 22 and 32 to 35 for discussions of opening and enclosure, interior and exterior.

		1 NOTES
	Epigraph
	'Froissart's Forge
	2	Raymond Queneau Exercices de style (Paris, Gallimard. 1947)
	3	Of course, the scientific approach of the Enlightenment sees Aristotle's historical hierarchy upturned -the
		universal subordinated to the particular by the tenets of empiricism.

4

public sphere : 5 ouvroir : 'workshop'. It is a more outmoded word than atelier (which is the one usually used for the 'writers' workshop'). It is cognate with oeuvre and Froissart's verb ouvrer. In fact the word was most often employed to refer to a specifically feminine place of work, usually in textiles (spinning or weaving). It has been revived in literary culture by the OuLiPo : l'Ouvroir de Littérature Potentielle, in which Raymond Queneau, Georges Perec and Italo Calvino have been the most celebrated spinners and weavers of text.

Madame of the Moment

6 point de repère, point d'appui, point de fixation, point de fuite... point... d'interrogation? 7 La princesse et le président, Valéry Giscard d'Estaing. (Paris, Editions de Fallois. 2009

  Your reserve… instead of putting me off, has whetted my appetite. I can see very well in this feigned prudishness the effect of your awareness of love' Splitting l'histoire15 It is a mark of his success in this venture that Voltaire is routinely, and wrongly, cited as the source of this idea of the siècle des lumiêres. As august a reference as Le Robert Dictionnaire historique de la langue française (1992, Holbach than Voltaire to be quite so straightforwardly graphic about an established religion 'paling and extinguishing itself before the luminaries of an enlightened century.' The Robert continues groping around for the glimmer of a source for this radiation by suggesting that the application of the term to the 'laïc' programme of scientific enquiry in the eighteenth century derives from a self-conscious decision to 'work according to the expression employed by Descartes': à la seule lumière naturelle. Amongst such flickering etymological shadows it is not immediately apparent whether it is the lexicographers or the lumières themselves who have chosen to overexpose Descarte's platonic metaphor by giving the word naturelle an atheistic sheen that he would never have allowed to see the light of day.'Il lut les lettres avec la même anxiété qu'il avait lu les Remarques. Le Siècle de Louis XIV, ouvrage plein d'agrément, de naturel et de facilité, avait été fait avec peu de lectures, peu de recherches, presque sans documents; on pouvait dire que Voltaire avait plutôt deviné qu'il n'avait étudié « ce grand siècle dont un souffle avait passé sur son berceau,» ["Sainte-Beuve Causeries du Lundi t.XIII"]. Il connaissait lui-même la faiblesse du fond, la ténuité des détails, et tremblait que La Beaumelle, avec les documents dont il faisait étalage depuis deux ans, ne vint à discréditer son chef-d'oeuvre.' (Lavallée 1865 vol 1. p.XVI) 18 '"... in a state of scanty undress, [she] had thrown herself into an armchair; there she thought at her leisure about her lover; often the day would find her sitting in a chair, her elbows leaning on a table, her gaze fixed in the ecstasy of love." Eh, my friend! did you see her in this state of scanty undress? Did you see her with her elbows on the table? Is it permissible to write history [the story] in this way?'

14 'Alain Rey et al.) sources the idiom thus : 'Cette dernière valeur, surtout au pluriel, les lumières (1665), a acquis une acception philosophique (1761, Voltaire : les lumières d'un siècle éclairé)' (vol. 1, p. 1152). In fact, this classic projection of a century of Enlightenment comes from the prefatory letter to the Recherches sur l'origine du Déspotisme Oriental attributed to Nicolas-Antoine Boulanger, but whose posthumous publication suggests very strongly that it (the preface if not the whole text) was actually the work of Paul d'Holbach. The context is enlightening : 'N'apercevra-t-elle point [la Police... 'the polity'] que la Raiſon & la Loi fondée ſur la Raiſon, doivent être les uniques Reines des mortels, & que lors qu'une Religion établie commence à pâlir & à s'éteindre devant les lumières d'un ſiècle éclairé, ce n'eſt plus qu'à cette Raison qu'il faut immédiatement recourir, pour maintener la Société,' (1761 p.XX-XXI) It would be much more like d'

  me de Maintenon au Roi. Février 1689. Saint-Cyr 21 Everything I was before Your Majesty unfolded me is written on this sheet. Your Majesty has broken the

vermilion of my most confidential seal. You hold, Sire, in your hands the whole of me; your fingers touch the creases of my trembling being. And yet, trembling, I am resolved to be opened out entirely by Your Majesty: to let you see, to let you read, what never should have been inscribed in ink or born in mind.

La Beaumelle had already published an unauthorised edition of this book in Frankfurt, with scathing marginalia.


Cédric Bentz
email: cedric.bentz@cnam.fr

Marie-Christine Costa
email: marie-christine.costa@ensta-paristech.fr

Pierre-Louis Poirion
email: pierre-louis.poirion@ensta-paristech.fr

Thomas Ridremont
email: thomas.ridremont@cnam.fr

Formulations for designing robust networks. An application to wind power collection

Keywords: Mixed-integer programming, bilevel programming, survivable networks

We are interested in the design of survivable capacitated rooted Steiner networks. Given a graph G = (V, E), capacity and cost functions on E, a root r, a subset T of V of terminals and an integer k, we search for a minimum cost subset E ′ ⊂ E, covering T and r, such that the network induced by E ′ is (k+1)-survivable: after the removal of any k edges, there still exists a feasible flow from r to T . We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set, a flow and a bi-level formulation where the second-level is a min-max problem (with an attacker and a defender). We propose algorithms for each problem formulation and compare their efficiency.

Introduction

Nowadays, the design of networks is crucial in many fields such as transport, telecommunications or energy. We are interested here in the design of survivable networks [START_REF] Kerivin | Design of survivable networks: A survey[END_REF][START_REF] Thomas | Strong formulations for network design problems with connectivity requirements[END_REF] able to route a given flow from a root to a set of terminals while respecting capacity constraints. We say that a network is (k + 1)-survivable if, after any breakdowns on k arcs, it is possible to route the same amount of flow. We also take into account the possibility of protecting a subset of k ′ arcs of the graph: those arcs cannot be deleted. The resolution of capacitated spanning and steiner tree problems has been studied in [START_REF] Jothi | Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design[END_REF][START_REF] Bentz | On the edge capacitated steiner tree problem[END_REF].

We introduce here the Capacitated Protected Rooted Survivable Network Problem (CPRSNP). Given a directed graph G = (V, A, c, u) where c and u are respectively the cost and the capacity functions on the set of arcs A, a set T ⊆ V of terminals, a root r ∈ V \ T and two positive integers k and k ′ with k + k ′ ≤ |A|, CPRSNP is to find a subset A ′ ⊆ A of selected arcs of minimum cost and a subset A ′ p ⊂ A ′ with |A ′ p | ≤ k ′ of protected arcs such that there is a feasible flow (i.e. it respects the arc capacities) routing a unit of flow from r to each vertex of T in the subgraph of G induced by A ′ , even if a breakdown occurs on any k arcs in A ′ \ A ′ p . We focus on wiring networks in windfarms, designed to route the energy produced by the wind turbines to the sub-station with respect to some technical constraints (cable capacities, non-splitting constraints, etc.) [START_REF] Hertz | Optimizing the design of a wind farm collection network[END_REF][START_REF] Fischetti | Proximity search heuristics for wind farm optimal layout[END_REF].Furthermore, we want those networks to be resilient to cable failures. The wind turbines are identical so we can assume that each one produces one unit of energy. Then A is the set of all possible cable locations and V = T ∪ J ∪ {r} where T (resp J) is the set of terminal nodes (resp. junction nodes between cables), and r ∈ V is the substation collecting the energy and delivering it to the electric distribution network. In that case, the flow is routed from T to r. In the case of windfarm wiring, one can see the protected arcs as robust cables or additional parallel cables. We consider a budget of protection such that we can only protect k ′ arcs.

We assume without loss of generality that u is an integer function. In this paper, we work on the oriented version of the problem, however the results can easily be adapted to the non-oriented one.

2 Formulations of the problem

Definitions and notations

We add to the input graph a vertex s connected to every terminal t ∈ T by a fictive arc (t, s) with c ts = 0 and u ts = 1. Then, s is added to V and the fictive arcs are added to A, and we denote by A I the set of initial arcs. Finding a flow which routes one unit of flow between r and each terminal in the input graph is equivalent to finding a flow of value |T | from r to s in the transformed graph. For each subset S ⊂ V , let δ -(S) be the set of arcs entering S. We have δ -(S) = {(i, j) ∈ A | i ∈ V \ S; j ∈ S}. We call Γ -(i) and Γ + (i) the sets of respectively predecessors and successors of a vertex i ∈ V .

Cut-set formulation

We introduce, for all (i, j) ∈ A, a binary variable y ij equal to 1 if and only if the arc (i, j) ∈ A ′ , and a binary variable p ij equal to 1 if and only if the arc

(i, j) ∈ A ′ p . Consider the r -s cuts [V \ V S , V S ] with V S ⊂ V , r ∈ V \ V S , s ∈ V S
and V S = {s}, and let S be the set of all the associated cut-sets in A, i.e. S = δ -(V S ). Notice that if S ∈ S then S ∩ A ′ is a cut-set in the selected network. For any set S ⊂ S, let C S k be the set of subsets of S of size k. We define M S as the maximum capacity of a subset of k selected and non-protected arcs of S:

M S = max C∈C S k (i,j)∈C u ij (y ij -p ij ) (1a)
M S corresponds to the maximum capacity that can be lost in the cut-set S after the deletion of k non-protected arcs. We propose the following cutset formulation:

min y,p (i,j)∈A c ij y ij s.t. (i,j)∈δ -(V S ) u ij y ij -M S ≥ |T | ∀S ∈ S (i,j)∈A p ij ≤ k ′ y ts = 1 ∀t ∈ T p ts = 0 ∀t ∈ T y ij , p ij ∈ {0, 1} ∀(i, j) ∈ A (2a) (2b) (2c) (2d)
Constraints (2a) ensure that for each cut, the capacity of the cut after the worst-case deletion of k non-protected arcs of the cut-set is at least equal to the number of terminals, i.e. the graph is (k + 1)-survivable. Constraints (2b) ensure that we have at most k ′ protected arcs while (2c) and (2d) guarantee that all fictive arcs are selected and none of them are protected. Notice that we do not have to impose p ≤ y since, from any optimal solution, we can get an equivalent one where this holds.

Constraints (2a) are non linear because of the use of the maximum operator in the definition of M S . To linearize it, we add the following constraints:

M S ≥ (i,j)∈C u ij (y ij -p ij ) ∀S ∈ S, ∀C ∈ C S k (3a)
The number of constraints (2a) and (3a) being exponential (and so is the number of variables M S ), we propose a constraints-and-columns generation algorithm. We begin with a small subset of S in constraints (2a) and (3a). We obtain a lower bound for our problem. Then we select a cut-set that does not verify some constraint (2a): given a network induced by the current value of y, we find by a using a MIP the cut of minimum residual capacity once we delete its k most capacitated non-protected arcs. If this capacity is inferior to |T |, we add the constraints associated to this cut-set, otherwise the algorithm terminates.

Flow formulation

Now we define F as the set of all possible arc-failure scenarios: it corresponds to the set of all k-combinations in A I . We introduce the variable x F ij which represents the amount of flow routed through the arc (i, j) ∈ A when the scenario F ∈ F occurs. The variables y and p are defined as in the previous formulation (Section 2.2). We propose the following flow formulation:

min x,y,p (i,j)∈A c ij y ij s.t. i∈Γ -(j) x F ij - k∈Γ + (j) x F jk = 0 ∀j ∈ V \ {r, s}, ∀F ∈ F t∈Γ -(s) x F ts = |T | ∀F ∈ F (i,j)∈A I p ij ≤ k ′ x F ij ≤ u ij y ij ∀(i, j) ∈ A, ∀F ∈ F x F ij ≤ u ij p ij ∀F ∈ F , ∀(i, j) ∈ F x ∈ R |A|×|F | + , y ∈ {0, 1} |A| , p ∈ {0, 1} |A I | (4a) (4b) (4c) (4d) (4e)
Constraints (4a) and (4b) ensure that there is a flow of value |T | for each arc-failure scenario. Constraint (4c) ensure that we have at most k ′ protected arcs. Constraints (4d) are the capacity constraints, for each scenario of failure. Finally, constraints (4e) ensure that in a scenario F where (i, j) ∈ F , we can route some flow through (i, j) only if the arc is protected. Notice that w.l.o.g. we can assume that p ≤ y.

The number of variables x F ij and constraints (4a) and (4b) being exponential, as in the previous section we propose a constraints-and-columns generation algorithm to solve the problem. We begin with a small subset of F . The separation problem is the problem of the k most vital links in a flow network [START_REF] Ratliff | Finding the n most vital links in flow networks[END_REF]: we search for the k non-protected arcs which, once simultaneously deleted, reduce the most the value of the maximum st flow.

Bilevel formulation

The bilevel formulation proposed here is particular in that the second-level is a min max problem. It can be seen as a game with a defender and an attacker (corresponding respectively to the leader and the follower).

For each (i, j) ∈ A, we introduce a variable x ij which corresponds to the amount of flow that the defender chooses to route through the arc (i, j). The variables y and p are defined as in Section 2.2. We also introduce the binary variables b ij , ∀(i, j) ∈ A: b ij = 1 if and only if the attacker chooses to delete the arc (i, j). We define the following polyhedron:

X (y, b, p) =          x ∈ R |A| + i∈Γ -(j) x ij - k∈Γ + (j) x jk = 0 ∀j ∈ V \ {r, s} x ij ≤ u ij y ij ∀(i, j) ∈ A x ij ≤ u ij (1 -b ij + p ij ) ∀(i, j) ∈ A I         
This polyhedron X (y, b, p) corresponds to the set of possible flows on the subgraph of G induced by the arcs (i, j) such that y ij = 1 provided they have not been deleted (thus non-protected). In X (y, b, p) there are the flow conservation constraints, the capacity constraints and the constraints imposing a flow equal to 0 on any arc which is non-protected and deleted. We also define the two following polyhedrons:

B = {b ∈ {0, 1} A | (i,j)∈A b ij ≤ k ; b ts = 0 ∀t ∈ T } Y = {(y, p) ∈ {0, 1} A 2 | p ≤ y ; (i,j)∈A p ij ≤ k ′ }
The polyhedron B defines the set of possible scenarios of arc failures while Y defines the set of possible selected and protected arcs (there are at most k ′ protected arcs in the selected network). Then we propose the following bilevel program:

min (y,p)∈Y (i,j)∈A c ij y ij s.t. f (y, p) ≥ |T | where f (y, p) = min b∈B max x∈X (y,b,p) j∈Γ + (r) x rj (5a) (5b)
At the upper-level, the defender selects the set of arcs he wants to add to the network as well as the one he wants to protect, by choosing a couple (y, p) in Y. The attacker then deletes some arcs by setting the variable b ∈ B in order to minimize the maximum flow that the defender will compute by setting the variable x in the flow polyhedron X (y, b, p). The aim of the defender is to ensure that this flow is at least equal to |T | (Constraint (5a)).

Consider the max problem in the lower-level: y, p and b are already fixed to b, p and ŷ. The problem is a max-flow problem from r to s, with two sets of capacity constraints. In our problem, the flow must be integer since it corresponds to a number of terminals. However, it is well known that the matrix of coefficients M in the arc-formulation of a max-flow is totally unimodular. Then, adding the second set of capacity constraints is equivalent to appending the identity matrix to M: the matrix remains totally unimodular and the capacities being integer, we ensure that the extreme points of the polyhedron defined by X (y, b, p) are integer. Thus, we can relax integrality constraints on x.

There always exists a feasible flow of value 0 and the lower-level problem is also trivially upper bounded. Hence, the strong duality holds and we can introduce the dual of the lower-level problem, after a slight reformulation due to the totally unimodular matrix: min

λ,µ,γ (i,j)∈A u ij ŷij λ ij + u ij (1 -bij + pij )γ ij s.t λ ij + γ ij -µ i + µ j ≥ 0 ∀(i, j) ∈ A µ r = 1 µ s = 0 λ, γ ∈ [0, 1] |A| , µ ∈ [0, 1] |V | (6a) (6b) (6c)
This problem is a special formulation of a min-cut problem: µ defines the two parts of the cut (sets of vertices i such that either µ i = 0 or µ i = 1). The variable γ ij is equal to 1 at least for all the edges (i, j) in the cut-set with p ij = 0 and b ij = 1, λ is equal to 1 for all other edges in the cut-set (if an edge (u, v) is not in the cut-set, we have λ uv = γ uv = 0). We denote the polyhedron defined by the dual constraints by D.

As the lower-level can be reformulated as a min min function by using the dual described above, it can then be rewritten as follows:

(2LP ) min b,λ,µ,γ (i,j)∈A u ij ŷij λ ij + u ij (1 -b ij + pij )γ ij s.t b ∈ B (λ, µ, γ) ∈ D
At this point, b is a variable so the objective function is non-linear. We linearize the terms b ij γ ij in a classical way by introducing binary variables l ij verifying the set of constraints

l ij ≤ b ij , l ij ≤ γ ij and l ij ≥ γ ij -(1 - b ij ), ∀(i, j) ∈ A,
which is denoted by L(b, γ). We also define the function g(y, p, λ, γ, l)

= (i,j)∈A u ij y ij λ ij + u ij γ ij -u ij l ij + u ij p ij γ ij .
s.t. b ∈ B l ∈ L(b, γ) (λ, µ, γ) ∈ D
In this bilevel reformulation, the polyhedron of the lower-level defined by B, L(b, γ) and D depends on neither y nor p. We can then consider the convex hull of the lower-level polyhedron and we denote by H the set of its extreme points; ( bh , λh , γh , μh , lh ) are respectively the values of (b, λ, γ, µ, l) at the extreme point h ∈ H. We can then reformulate the bilevel formulation as a single-level one as follows:

min (i,j)∈A c ij y ij s.t. g(y, p, λh , γh , lh ) ≥ |T | ∀h ∈ H (BP) b ∈ B l ∈ L(b, γ) (y, p) ∈ Y (λ, µ, γ) ∈ D (7a) (7b) (7c) (7d) (7e)
Constraints (7a) ensure that for each extreme point of H, f (y, p) is greater than |T | (i.e. the minimum value of f (y, p) over the polyhedron defined by (7b), (7c) and (7e) is greater than |T |), meaning that the flow cannot be decreased after any k breakdowns.

Remark 2.1 In (BP), g(y, p, λ, γ, l) is non-linear because of the products y ij λ ij and p ij γ ij but they can be linearized as it has been done for b ij γ ij above.

However, there is an exponential number of constraints (7a) and we do not know how to describe explicitly the convex hull To tackle this issue, we use a constraints generation algorithm where we relax the set of constraints (7a) and use (2LP ) as the separation problem: while the optimum value of (2LP ) is lower than |T | for the current optimal solution (ŷ, p), we generate the constraint (7a) associated to the extreme point corresponding to the optimal values of (b, λ, γ, µ, l) in (2LP ).

Proposition 2.1 Let (ŷ 1 , p1 ) and (ŷ 2 , p2 ) be two feasible solutions of (BP) such that ŷ1 ≥ ŷ2 and p1 ≥ p2 . If adding a constraint g(y, p, λ, γ, l) ≤ g(y, p, λa , γa , la ) makes any solution with (y, p) = (ŷ 1 , p1 ) infeasible, then it also makes any solution with (y, p) = (ŷ 2 , p2 ) infeasible.

Proof. For any value ( λa , γa , la ) of (λ, γ, l), we have g(ŷ 1 , p1 , λa , γa , la ) ≥ g(ŷ 2 , p2 λa , γa , la ) since ŷ1 ≥ ŷ2 and p1 ≥ p2 (as u, λ and γ are positive). Hence, if g(ŷ 1 , p1 , λa , γa , la ) ≤ |T | -1 then g(ŷ 2 , p2 , λ S , γ S , l S ) ≤ |T | -1. ✷ To improve the cut obtained by solving (2LP ), we try to inject better values ŷ of variables y in it. To get these values, we first solve the following problem, and then we compute the new ŷ accordingly (as explained later). Given a starting solution (ŷ, p), we want to find a cut-set in the support network with a minimum number of arcs such that this cut-set is non-valid in the network induced by (ŷ, p) (meaning that if we remove k non-protected arcs of the cut-set, its remaining capacity is smaller than |T |). This can be modeled as follows:

min (i,j)∈A λ ij s.t (i,j)∈A u ij ŷij λ ij + pij γ ij ≤ |T | -1 (i,j)∈A γ ij ≤ k γ ts = 0 ∀t ∈ T (λ, µ, γ) ∈ D µ ∈ {0, 1} V (8a) (8b) (8c)
The variables (λ, µ, γ) define a cut as in (2LP ) since they belong to D (recall that D is the set constraints ( 6) ). However, adding the other constraints makes the constraints matrix not unimodular anymore: we have to set µ as a 0-1 variable. Constraint (8a) ensures that the cut-set selected is non-valid (as defined before). Constraint (8b) bounds the number of deleted arcs to at most k, while constraints (8c) forbid the deletion of fictive arcs.

Then the new values of ŷij are computed as follows: we set ŷij to 1 for all (i, j) with λ ij = γ ij = 0 and let the others to their current value. It implies that the new value of y will include the previous one and, using Proposition 2.1, we generate a better constraint than the original by calculating the extreme points associated to this new value of y.

Tests and conclusion

All experiments were performed on a computer with a 2.40GHz Intel(R) Core(TM) i7-5500U CPU and a 16GB RAM using the solver CPLEX version 12.6.1. We present 3 tables of results. For each table, we present the solving time (limited to 2000 seconds) and the gap if the optimum is not reached. The cut-set and flow formulations were solved using an iterative procedure in which after the generation of the constraint, the model is resolved as a MIP.

Table 1 shows the performances of the different formulation on an uniform capacities generated instance (those results reflect the general tendency obtained on tests made on other instances) with no protected arcs. One can see that the cut-set formulation is the most efficient one: this can be explained by the fact that the variable M S is a constant in this case. The bilevel formulation is also efficient here whereas the flow formulation becomes inefficient for k ≥ 2. Table 2 shows the difference of performance between the three different formulations for different values of k and k ′ on a generated instance with nonuniform capacities. Although the flow formulation competes with the bilevel one for k = 1, it seems obvious that the bilevel formulation is the best one to solve CPRSNP when k > 1. This can be explained by the fact that the number of variables (M S and x) and constraints is exponential in k for the two other formulations.

Table 3 shows the results obtained on three different instances of the bilevel formulation for different values of k and k ′ . We can see that even if the solving time generally grows with k and k ′ , the formulation is much less sensitive to those two parameters.

  We can then rewrite the bilevel program as: min (y,p)∈Y (i,j)∈A c ij y ij s.t. f (y, p) ≥ |T | where f (y, p) = min b,λ,γ,µ,l g(y, p, λ, γ, l)

Table 1 :

 1 Comparison of the performance for uniform capacities with k ′ = 0

	Instance	Bilevel	Cut-set	Flow
	|V |-|T |-|A| k k' t (s) gap t (s) gap t (s) gap
	20-5-90	1 0 5.9	0	2.17	0	21.3	0
	-	2 0 49.3	0	7.8	0	2000 0.25
	-	3 0 22.8	0	21.1	0	2000 0.15

Table 2 :

 2 Comparison of the 3 formulations for |V | = 30, |T | = 3, |A| = 140

	I	Bilevel	Cut-set	Flow		Instance		k ′ = 0		k ′ = 1	k ′ = 2		k ′ = 3
	k k' t (s) gap t (s)	gap t (s) gap	|V |-|T |-|A| k t (s) gap t (s)	gap t (s) gap	t (s)	gap
	1 0 9.3	0	1812.2	0	15.8	0	20-5-100	1	6.4	0	19.0	0	26.5	0	18.0	0
	1 1 11.6	0	658.8	0	5.14	0	-	2 15.2	0	58.9	0	167.2	0	223.7	0
	1 2 13.0	0	691.3	0	5.3	0	-	3	-	-	58.2	0	315.8	0	822.3	0
	2 0 43.8	0	2000 0.12 2000 0.01	25-8-120	1 22.0	0	152.0	0	87.9	0	67.3	0
	2 1 30.9	0	2000 0.22 2000 0.05	-	2 76.4	0	161.1	0	546.3	0	794.4	0
	2 2 30.5	0	2000 0.04 112.6	0	-	3	-	-	34.2	0	158.0	0	1670.3	0
	3 0 29.0	0	2000 0.19 2000 0.18	35-3-175	1 35.7	0	101.4	0	82.3	0	159.6	0
	3 1 21.0	0	2000 0.24 2000 0.21	-	2 180.6	0	1077.3	0	2000 0.11 1245.7	0
	3 2 66.5	0	2000	0.2 2000 0.16	-	3 244.3	0	2000 0.03 2000 0.45 2000 0.49

Table 3 :

 3 Performance of the bilevel formulation