Nathalie Ferrand

Putting Oneself in the Light: Rousseau and the Frontispieces of His Works Technically speaking, the engraving is a play with shadow and light. Diderot pays homage to it in just such terms in his Salon de 1765: "Regardez chaque point comme un rayon de lumière éteint. [...] Dans la taille-douce tout est éclairé, le travail introduit l'ombre et la nuit.

Dans la gravure noire la nuit est profonde; le travail fait poindre le jour dans cette nuit" 1 .

Jean-Jacques Rousseau was so enamoured of the colourless world of the engraving and of its intense expressiveness that hardly a single work of his was given to the world unaccompanied by some engraving. He was, moreover, himself a collector of such prints.

Nor was it just in those works of his that fall into the category of the "literary" that Rousseau called the printed image to his aid. He did the same in his philosophical publications, placing at the threshold of each published work vignettes and frontispieces whose conception was the fruit of original reflection on the relation between the text in question and the signifying force of the engraved print. He would himself take the decision as to just which scenes and topics were to be illustrated and would even actively supervise the work of the draughtsmen and engravers, often intervening to regulate, so to speak, the luminous intensity of the images. Such was the case, for example, of the tenth engraving in La Nouvelle Héloïse, which represents St. Preux's nightmare (Ill. 1) 2 . Rousseau, we know, several times requested that this plate, already dark in its initial version, be made still darker so as to create an atmosphere of terror: "The proof I have seen of this plate with the ghosts will do very well, provided it is made still darker and that a slightly more alarmed expression is given to the face"; "the 'ghosts' print will be an admirable one, once its lighter areas have been filled in and darkened" 3 . We find him making an analogous, even if photically inverse, intervention with regard to the first of the engravings which provide the illustrative plates for Emile, 1 Denis Diderot, Salon de 1765, in:

complaining that the flames of Hades had not been made bright enough in the scene of Thetis dipping her infant son Achilles into the waters of the Styx (Ill. 2) 4 .

What the present essay sets out to examine is how the engravings thematize and address, in such cases, the same concerns as are at issue in the substantial conceptual content of the respective texts themselvesthat is to say, how engravings direct the luminous flux , as it were, of the reader's attention toward this or that essential aspect of these texts, sometimes hidden away in these latter's obscurest corners. I am going to begin with a rapid overview, in terms of a few broadly drawn chronological stages5 , of the principal engravings which accompanied Rousseau's texts from 1751 on. I am then going to focus in on one particular case among these engravings: one which exemplifies especially clearly the connection between light and image in the engraver's art while at the same time demonstrating the conceptual value of images in Rousseau's work.

Rousseau and the Illustration of His Texts from 1751 to 1764

The first edition of the Discourse on the Arts and Sciences, in 1751 (Ill. 3) represents an inaugural moment in this regard, with a frontispiece that can be said to be foundational for Rousseau's work as a whole. This is so inasmuch as the frontispiece gives graphic expression to the metaphor of Enlightenment but also, as we shall see below, to that of Creation. Its theme is a reprise of the mythological one of Prometheus giving fire to Man, with a satyr shown advancing from the side trying to grab the gift before it is passed on. 1751 is also the year which sees the start of the publication of the Encyclopaedia which, very significantly, likewise adopts, in the vignette which appears on the title page of its first volume, the metaphor of "Enlightenment illuminating the world" expressed via the motif of a winged figure, bearing a flame on his forehead, advancing through parting clouds. If the Encyclopaedia's vignette proposes a triumphalist interpretation of this theme -its interpretation, as it were, in a major keythe Rousseauan frontispiece interprets this same theme, as we shall see below, in a minor key, a more critical one, since the satyr serves to evoke a certain mistrust of the knowledge symbolized by Prometheus's flaming torch. The drawing is attributed to the painter Jean-Baptiste Marie Pierre, although it was then engraved by Jean-Charles Baquoy, whose name was the only one to appear below the actual plate.

Well-practiced in the treatment of mythological themes, Pierre was a great historical painter whose career can stand almost as a microcosm of the whole history of 18 th -century French painting. He did not, for all that, consider himself above the illustration of the writings of his contemporaries, so that his works include, besides his many paintings in the traditional canvas form, also a significant number of such illustrative engraved drawings 6 .

In 1755, on publication of Rousseau's second discourse, the Discourse on the Origin of Inequality, the philosopher's readers found themselves presented with two modes of literary illustration at once: namely, on the one hand a frontispiece and on the other a vignette placed in the middle of the title page presenting an allegory of liberty. As is indicated by the caption beneath the frontispiece, the theme which this latter illustrates is drawn from one of the Discourse's footnotes: "See note 13 p. 259". This type of reference, quite unusual in nature, had already been practiced in the frontispiece of 1751: "See note pag.31". The practice of the illustration of footnotes, of which we see an instance here, might well be a form of erudition interesting to study in its own right, so as to broaden somewhat the spectrum of phenomena already analysed by Anthony Grafton in his The Footnote: A Curious History 7 . In any case, illustrations of this unusual type appear at the threshold of each of Rousseau's first two Discourses. Perhaps there is something slightly provocative in thus drawing the reader's attention to what seems to be a mere detail lodged within the infra-paginal space of the text.

But such a practice is in fact indicative of how demanding Rousseau is of his readers, requiring of them a constant close attentiveness to what and how he writes. The experience of this illustrated footnote allows one to perceive the contrast in rhythm between the rapidity of the perception of the image and the slowness of a true, attentive reading. Because in order truly to understand just what is being referred to here, one must have the patience to read through an entire text in all its complexity. What is illustrated in the frontispieces both of the first and of the second of Rousseau's Discourses is notor is not solely -Rousseau's respective individual text. It is rather, in an important sense, the entire personal library on which Rousseau was drawing for each of these. In the frontispiece of the first Discourse, the 6 Nicolas Lesur/Olivier Aaron, Jean-Baptiste Marie Pierre, 1714-1789: premier peintre du roi, Paris 2009. This work comprises a list of engravings attributed to Pierre but the engraving in question here does not feature on it, since it was not signed by the artist. 7 Anthony Grafton, The Footnote. A Curious History, Cambridge, Mass. 1997.

reference is a classical one, specifically to Plutarch (see below); in the second, it is to an anecdote drawn from an early modern travel narrative describing the failure of the attempts by European explorers to civilize and convert to Western values the savages encountered in remote regions of the world, these latter proving to be much more passionately attached to their own origins and customs than the Europeans had expected. That is to say, after the visualization in the first frontispiece of the failure of culture, the second visualizes the failure of the civilizing process.

In 1761 and 1762, for La Nouvelle Héloïse and Emile, Rousseau opts for the technique of engravings arranged in series, thus prompting readers to observe a certain highly specific rhythm in their reading of these lengthy and substantial works. When La Nouvelle Héloïse is illustrated for the first time in 1761, this work of illustration comprises no frontispiece, but rather consists of twelve plates inserted at strategic points in the story's development.

Historically, it was in Germany, in the translation by Gellius which also appeared in 1761, that La Nouvelle Héloïse was to be provided, for the first time, with a frontispiece, depicting a scene of reading bringing together St. Preux and Julie 8 . In Emile, on the other hand, the series of engravings consists only of five frontispieces (all mythologically-themed) which open, respectively, the five books that together make up the whole work.

The Social Contract, first appearing in 1762, features a vignette on its title page, the allegorical significance of which slipped and shifted somewhat throughout the book's earliest editions. This was due to certain miscommunications between Rousseau and his publisher, Marc-Michel Rey. Rousseau had initially agreed that the allegorical figure representing liberty which had been used seven years earlier on the title page of the Discourse on the Origin of Inequality could be used once again on the title page of The Social Contract. He had stipulated, however, that this figure, whom he humorously referred to as "chubby-cheeked Liberty", should be redrawn, for the new title page, in a rather more dignified style. In the first of the two printings of the text which he oversaw in this year of 1762, Rey had the figure revised into a standing personage whose traits, however, turned out, due to a confusion about the symbolic attributes conventionally assigned to such personifications, to be, strictly speaking, more an allegorical representation of Justice than one of Liberty. Rousseau expressed his dissatisfaction with this and, in the second printing of The Social Contract, Rey had the image revised once again in a way that realigned it somewhat with the original 8 Nathalie Ferrand, Livres vus, livres lus. Une traversée du roman illustré des Lumières, Oxford 2009, p. 66-67.

vignette from the Discourse on the Origin of Inequality. That the same iconographic motifs and symbols were taken up in both publications establishes a certain continuity between these two political works of Rousseau's, and the placing of such an allegory at the threshold of both texts shows how right E. Cassirer was to see in Rousseau above all a thinker of liberty.

1764 is a pivotal moment: namely, that of the appearance of a first full edition of Rousseau's Works under the imprint of the Parisian publisher Duchesne. In this year, Duchesne assembles, in nine volumes, all the works of Rousseau published up to that point and enriches them with a series of engraved plates, the first of which (Ill. Consequently, it takes on great importance. This edition provided the opportunity for certain of Rousseau's texts to be illustrated for the first time. Such was the case, for example, of Le Devin du village, or of the Letter to d'Alembert (the latter with a frontispiece showing the Judgment of Ares). La Nouvelle Héloïse, in this edition, is provided with an allegorical frontispiece by Cochin.

The 1764 frontispiece of the Discourse on the Arts and Sciences thus becomes the emblem of an entire oeuvre and perhaps the emblem of the characteristic self-positioning of this oeuvre's author. This is why it is imperative to linger over the details of this frontispiece, taking care to examine the genesis of its design, its content, and its various significations 9 .

Prometheus and the Satyr: Genesis of a Frontispiece

That we are able to consider at all the 1764 frontispiece of the Discourse on the Arts and Sciences from the point of view of the genesis of its design is due to the exceptional circumstance that there have come down to us, intact, several versions of this plate produced in the course of the realization of the final printed illustration. These consist in two preliminary sketches, one in pierre noire pencil (Ill. 4) and one in ink (Ill. 5), and two artifacts marking stages in the actual process of engraving: namely, a proof etching "before letters" 9 The following remarks represent a concise summary of the argument of a longer study which I have published under the title Variation sur Prométhée et le satyre: la genèse du frontispice de Gravelot pour le Discours sur les sciences et les arts, in: Genesis 50 (2020), p. 167-175. and a copy of the work in something close to its final state, the etching having here been supplemented by burin-work creating shadow and relief. There appear beneath the more fully completed etching the signatures of both engraver and draughtsman. It is probable, then, that it represents an initial, tentative printing of the completed plate itself.

Gravelot's drawings are currently held in the Houghton Library at Harvard 10 , and the various artifacts testifying to the different stages of the engraving work carried out by Joseph de Longueil are to be found in a private collection in Britain, preserved as part of the famous portfolio which was given as a gift by Rousseau to George Simon Harcourt at the time of Rousseau's leaving that country and which contains proof copies of the illustrations to his own works that the author liked to collect 11 where Rousseau sets about showing that "the sciences and the arts owe […] their birth to our vices." 12 . Rousseau then evokes various myths of antiquity in order to show that both the Greeks and the Egyptians early on warned the human race against the danger posed by the arts and the sciences. He mentions in particular the myth of Prometheus, chained to his rock as punishment for having brought fire to humankind. A note here adds:

It is easy to understand the meaning in the Prometheus fable. It does not seem that the Greeks, who chained Prometheus to the Caucasus, were any better disposed toward him than were the Egyptians toward their god Theutus. 'The satyr, according to an ancient fable, wanted to kiss and embrace fire, the first time he saw it. But Prometheus cried out to him, Satyr, you will mourn the beard on your chin, for it burns when you touch it.' This is the subject of the frontispiece. 13 The quotation inserted into the note comes directly from the Moralia of Plutarch, who had, in his turn, borrowed this burlesque scene from a "satyr-drama" (now lost) of Aeschylus 10 The present study was written after a visit to this institution to examine them on site. 11 On this portfolio see Ann-Marie Thornton, A Gift from Jean-Jacques Rousseau to George Simon Harcourt: Etchings and Proofs of the Illustrations to His Works, in: Eigtheenth Century Fiction (April/July 2002), p. 441-463. The portfolio remains the property of the descendants and heirs of G. S. Harcourt, whom I thank for their permission to photograph its contents. 12 Jean-Jacques Rousseau, Discours sur les sciences et les arts, in OEuvres complètes. Vol. III, Paris 1964, p. 17. 13 Jean-Jacques Rousseau, The Social Contract and the First and Second Discourses, ed. a. with an introduction by Susan Dunn, New Haven/London 2002, p. 56 (translation slightly amended).

(Prometheus the Fire-Lighter). We see here, then, a "nesting" not just of one text within another but of both within yet a third. In Aeschylus, a troop of satyrs had wanted to steal the fire that Prometheus was bringing to humankind, but burned their beards in the attempt 14 . This scene, merely ludic in Aeschylus's treatment of it, had been endowed with moral significance by Plutarch, who used it to point up how fire was useful despite its dangerousness 15 , and was now made, by Rousseau, to bear yet another meaning: that of a warning against the dangers posed by the arts and sciences. The figure in danger in this last Rousseauan treatment of the theme is a naïve and over-inquisitive satyr, who represents a mankind still at a primitive stage of development and needing to be protected from the moral corruption which will tend to be brought about by the sciences and the arts. In this transferral, then, from one author to another, there is to be observed a change in the whole meaning of the scene. And in the transferral into the sphere of the image, still other significant alterations occur. Thus, in Rousseau's text, the situation evoked is one involving two actors only (Prometheus and a satyr); in this textual passage's graphic interpretation, however, the initial illustrator Pierre (Ill. 2) adds a figure in the centre, making of it a scene involving three actors, an alteration which Gravelot then retained.

Rousseau's choice of Prometheus as the hero of the frontispiece of the Discourse on the Arts and Sciencessurprising though it wasprovided a fine opportunity for a painter such as Pierre to combine together several component parts of the myth, and several different aspects of the figure of Prometheus 16 coming to light in his existence prior to his punishment by the gods. For the crafty thief, who proved capable even of deceiving Zeus 17 , was also 14 Aeschylus's play, only a few fragmentary verses of which have come down to us, followed the rules and conventions described as those of the "satyr-drama" by François Lissarague in his La Cité des satyres. Une anthropologie ludique (Athènes, vie-ve siècle avant J.-C.), Paris 2013, p. 24: "These dramas represented a sort of 'playing with myth' consisting in superimposing, upon some given history well-known to the entire audience, a group of satyrs who then proceeded to react to the given situation in a manner typical of satyrs. The recipe, in short, would be the following: take a well-known mythical episode, add in satyrs, mix well, and stand back and observe the result." 15 Les oeuvres morales et philosophiques de Plutarque, translatées de grec en françois, par Messire Jacques Amyot, Paris 1618, p. 109 of the chapter How One May Acquire Some Benefit from One's Enemies. The anecdote is cut short by Rousseau and acquires a meaning contrary to that which it had borne in Plutarch who had, in the end, assigned a positive value to fire, pointing out that "although fire burns when one touches it, it is also the procurer of light and heat and can be, once one has learned to make proper use of it, the useful agent of all forms of art. Similarly, one's enemy [etc.]". See Plutarque OEuvres morales, Paris 1989. Vol. 1, 2nd part, p. 197. 16 On the complexity of the literary figure Prometheus, see the book by Raymond Trousson, Le Thème de Prométhée dans la littérature européenne, Geneva 1964 (republished in 2001). 17 Far-sighted Prometheus represents "the very type of that devious intelligence, that power of deceit, which the Greeks designated by the name 'mètis'"; Prometheus "knows in advance exactly what is going to happen": Marcel Detienne/Jean-Pierre Vernant, Les ruses de l'intelligence. La mètis des Grecs, Paris 1974, p. 63. renowned in myth as an artist18 and remembered too, in a tradition which was preserved and frequently artistically represented still in the 18 th century, as the creator of Man himself19 .

In the 1751 frontispiece, Prometheus occupies the uppermost part of the image. Larger than the other figures, he is borne upon a cloud, indicating his belonging to the world of the gods20 . His forehead encircled by a headband, and bearded as was iconographically traditional for this figure,21 , he holds in his right hand a torch which he is placing above the head of a statue of a beautiful ephebe; his left hand is resting on the shoulder of this not-yet-animated human statue in a Pygmalionesque gesture, which suggests that this touch is about to infuse life into an as yet merely sculptural being. This latter beingthe one human and mortal (even

if not yet animate) figure in the groupis placed much lower in the image, but at its horizontal centre, fully visible and illuminated and with both his feet on a pedestal. At the moment captured in the image, this figure is the least alive of the three, even if it is clear that his presently empty gaze is about to light up and his body, still frozen just at present in an attitude borrowed from the codes of antique statuary, is about to acquire life and motion under the effect of the fire stolen for his sake from the Olympians. The addition of this sculptural figure to the composition may be understood as resulting from the need to provide an allegorical personification of the arts, Prometheus's torch symbolizing rather the sciences, or knowledge in general.

Lowest down in the image, we see a third actor, slightly hidden in shadow and somewhat in the background. This is the satyr, who appears to be rushing into the image from outside it, with only one hoof resting on the ground and his arm raised high into the air as if in an attempt to seize the fire of the torch, to the dangers of which he is clearly entirely oblivious.

Half-human and half-animal, he represents humanity at the most primitive stage of its development, a being still immersed in Nature, a "wild man". The trajectories of the respective gazes of Prometheus and the satyr together form an oblique line which transects a second oblique line formed by the long cloud on which Prometheus is borne and does so at a point which constitutes the central symbolic element of the image: the flame of the torch, which brings life and knowledge and which is the point at which the metaphor of "Enlightenment" finds its concentrated embodiment. The caption below the engraving is a highly enigmatic one for anyone glancing at it before having read the Discourse itself: "Satyr, you do not know it. See note page 31." The inscription of these words below the plate accentuates the presence of the satyr in the image. He becomes the subject addressed by it, addressed by the word of Prometheus, just as the reader is the subject addressed by the word of Rousseau.

The frontispiece serves, indeed, to present the work to its "target audience" and to signal some of the key topics and issues at stake within it. But it also constitutes, whether intentionally or not, a way of presenting an author. This is quite especially the case within this specific context of a prize set for public competition by an Academy: a context in which the winner of such a prize necessarily finds himself exposed to public scrutiny. One might well, then, pose the following question: Where is Rousseau himself to be situated within the set of permutable roles played out in the scene represented in this frontispiece? Is he Prometheus? In a sense he certainly is because, like Prometheus, Rousseau aspires to enlighten mankind and delivers, as a "prophet of truth" (J. Roger 22 ), his message to this latter in the same tonea tone at once stern and intimateas Prometheus adopts when speaking to the unknowing satyr (the "you" of the "you do not know it, satyr" is in fact the informal, amicable "tu"). Like Prometheus too, Rousseau was later to attempt (in Émile) to form and mould a new type of human being anda third and final common trait between themwas to end up a victim of his own audacity. In a way, then, this opening engraving is itself prophetic. But the roles are unstable. Might Rousseau's role not possibly also be that of the human statue which appears about to take on form and life here at the opening of his book? This too might very well be the case in a symbolic sense, because every writer is the product of his or her own books. This applies quite especially to the book here in question, since it was this Discourse on the Arts and Sciences that gave birth to Rousseau as a public, recognized author 23 . Indeed, we find 22 "From this point on there has, once and for all, been constituted the image of the 'pedagogue of the human race', the prophet of truth illuminated by the sudden light of Revelation" writes Jacques Roger in his introduction to the Discours sur les sciences et les arts, Paris 1992 [1971], p. 15. 23 In a "notice to the reader", which was written later in life and intended for a re-edition of the first Discourse, but which remained, in the end, just an unpublished manuscript, Rousseau was to write: "What is fame? Here Rousseau himself recounting, in a famous letter to M. de Malesherbes, his birth as a writer at the moment of composing the first of his Discourses in terms (those of the so-called "illumination of Vincennes") which closely resemble what we see in the frontispiece. The "enlightened" statue portrayed in the engraving could have said of itselfand so might have run a caption for the plate alternative to the actually printed legend: satyre, tu ne le connois pas -"all of a sudden I felt my mind dazzled by a thousand lights"24 . One might go so far as to say that, placed as it is at the threshold of the first Discourse, the engraving implicitly adds to the theme already announced by the title and by the terms of the competition the equally essential theme of creation: the creation at once of the human species (symbolized by the statue that comes, in a Pygmalionesque moment, to life), of an author, and of a body of work.

One might even ask, indeed, whether one might not legitimately adopt yet a third interpretative point of view and understand Rousseau's position within the image to be that of the satyr, this latter being read as a being "according to Nature", bearer of the values of the human species in this species' earliest form. The plate, however, appears to rule out such a hypothesis, since, in its caption, the satyr is addressed as a "you", whereas the author is necessarily and inevitably an "I". Set back a little, situated in shadow and even, in part, externally to the frame of the image, the satyr occupies a decidedly marginal position. This matches, indeed, his status as a hybrid creature at the margins of humanity; but it could also be construed as expressing a certain affinity or proximity of the satyr to Rousseau who also presents himself, on the title page, as a marginal being, namely in the form of a misunderstood "barbarian"25 . Pierre, in any case, chose to visually "sideline" the figure of the satyr, leaving to Gravelot, as we shall see, the possibility of proposing another possible way of proceeding.

Let us turn, then, to the work of Gravelot, which can be examined in the two drawings held in the Houghton Library, both drawn from a large portfolio of sketches containing at least 1,800 drawings, now scattered in collections all around the world 26 . They give us access to the is the unfortunate work to which I owe mine. It is certain that this piece [...] made my name [etc.]": Rousseau, Correspondance, p. 1237. creative universe of this artist who collaborated with the most important writers of the 18 th century. A painter by training, but specializing in engraving, Gravelot represented that new profession which did not, at this time, yet adopt the name "illustrator", but certainly underwent considerable expansion and development in step with the growth of the bookselling business in the 18 th century. 27 Gravelot was perhaps the best-known practitioner of this as-yet-nameless profession. He had already collaborated with Rousseau on the engravings for La Nouvelle Héloïse (1761), an experience that has passed into critical history as an unenviable one for the draughtsman who, we are told, suffered much from the demanding, indeed unrealizable stipulations of the writer28 . Gravelot, however, proved able to navigate the perils of this collaboration, revealing all the depth of Rousseau's discourse by contributing an illustrative vision of his own, which was creative while remaining closely pertinent to the text. At the time of the re-edition of the Discourse under the imprint of the publisher Duchesne in 1764, when his frontispiece was called upon to decorate the first volume of the OEuvres de M. Rousseau,29 Gravelot was working, indeed, within a constraining framework, since the text with which he was dealing had already previously been illustrated and he was obliged to comply both with the author's instructions and with the existing graphic realities of the preceding frontispiece, which had already come to form part and parcel of the history of the text. What margin of manoeuvre, then, as regards possible variations did Gravelot enjoy30 ?

The alterations vis-à-vis the 1751 frontispiece which we observe in the preparatory pierre noire pencil sketch made for its 1764 successor (Ill. 4) are focussed principally on the figure there is to be found, in the form of sketches, initial ideas, studies and graphic emendations, pretty much the entire oeuvre of Gravelot, amounts in total to around 1,800 sheets and is of the greatest possible interest […] [It] comes from the Marquis de Fourquevaulx", see Roger Portalis, Les dessinateurs d'illustrations au dixhuitième siècle, Paris 1877, Vol. I, p. 293-294. In the possession, at the time, of the collector Emmanuel Bocher, the drawings that originally belonged to it became scattered in the course of the 20 th century. A certain number of them are today to be found in US libraries.

his "wildness": Gravelot gave to him, via the discreet play of accessories added as parts of the décor, the attributes of the god Pan, god of pastoral culture and a musical god, namely, the shepherd's crook and pipes of the style that Pan was supposed to have invented 35 . In the course of his work on the engraving, then, Gravelot can be said to have subtly shifted the accent of this scene: he has discreetly raised the satyr to the rank of a divinity and has transported the action into Arcadia, habitat of the god of shepherds and wooded groves. The violence is allayed. We are not far removed from an atmosphere of bucolic daydreaming anticipating the way in which Rousseau's mental and moral world came to be perceived after the performance of Le Devin du village and the publication of La Nouvelle Héloïse.

Conclusion

What did Rousseau think of the work? His response to his publisher, who sent him the etching in the summer of 1763, was a stinging one: "the Prometheus [proof] has all the faults of M.

Pierre's but none of its beauties, such as the figure of Prometheus himself. One cannot distinguish whether the satyr is making to embrace the fire or the statue, a highly inappropriate ambiguity which it would have been easy to avoid just by placing the torch on the side of the satyr so that it would be almost up against his face" 36 . But in order to satisfy this objection of Rousseau's, it would have been necessary to shatter the equilibrium which Gravelot had succeeded in finding and to restore to Prometheus that foremost rank among the image's figures which he had forfeited, becoming somewhat evanescent, in this new version. This was not done and the frontispiece was printed without further modifications.

It was Gravelot, then, who, against Rousseau's own expressed desire, succeeded in bringing out the centrality both to the Discours and to Rousseau's whole oeuvre of this liminal figure of the satyr: a disobedience to which we owe, perhaps, the possibility of a political reading of this frontispiece. Decades later, one of the most prominent antagonists of the French Revolution, Antoine de Rivarol, was to write, as a reproach cast against Rousseau for not having sounded a clearer warning against (what Rivarol perceived to have been) the misdeeds of the Enlightenment, the following lines, which show how profoundly Rivarol had 35 These pipes and the shepherd's crook are the attributes of the god Pan according to Pierre Grimal, Dictionnaire de la mythologie grecque et romaine, Paris 1951, p. 342. The preferred instrument of the satyr was the type of pipes known as aulos. 36 Letter of 21 st August 1763, to Nicolas Bonaventure Duchesne. himself been marked by the frontispiece of the first Discourse and how far the satyr (i.e. the people) approaching the fire (i.e. knowledge and power) in order to seize it had, all along, really been the matter most essentially at stake in this frontispiece: "Rousseau had engraved at the head of his political works a satyr reaching out for a torch, and he cries to him: Satyr, stand back, for fire burns. But he explained his allegory here very badly, because the satyr, being far away still, is struck only by the torch's light. The warning cried out to him, then, ought to have been Satyr, stand back, for light burns. And this, indeed, was the heart of the matter. Our philosophers threw the light to our satyrs, without considering the fact that light burns" 37 . But it was too late. The satyrs had seized the fire and learned to use it.

  3), signed by Gravelot, is a new version of the frontispiece of the Discourse on the Arts and Sciences. This opening image thus plays a double role: it both serves as a frontispiece to the first volume of this set of Works (the one containing the Discourse on the Arts and Sciences itself) and forms a visual threshold for the entire body of work published by Rousseau up to that point.

  . The passage illustrated by Gravelot (and by J.-B. M. Pierre in his first version of the frontispiece from 1751) is one located right at the start of the second part of the Discourse

  Translated into English by Alexander Reynolds 37 Pensées inédites de Rivarol, 1836, pp. 129-130, see Roland Mortier, Clartés et ombres du siècle des Lumières: études sur le XVIIIe siècle littéraire, Geneva 1969, p. 51-52.

Letter to Duchesne and Guy dated 7 th of March 1762 (CC 1704).

For a full panorama of the images to which I will refer in the present essay the reader is referred to Jean-Jacques Rousseau et les arts. Catalogue of the exhibition held at the Pantheon, 29 June -30 September 2012 under the direction of Guilhem Scherf, Paris 2012, in particular to p. 54-91, "L'illustration des oeuvres de Rousseau" by Nathalie Ferrand. Also worth consulting is the pioneering catalogue of the exhibition Jean-Jacques Rousseau face aux arts visuels. Du premier Discours au rousseauisme. Neuchâtel, 20 September -23 November 2001 under the direction of Cecilia Hurley, Neuchâtel 2001.

As is shown, for example, by the series of paintings by Piero di Cosimo, see D. Arasse, Le Sujet dans le tableau, Paris 2006, p. 110-113.

In the graphic arts, for example, a painting by Louis de Silvestre exhibited in 1702 at the Académie royale de peinture et de sculpture showed The Forming of Man by Prometheus, Aided by Minerva (preserved today, together with the sketch that it was based on, at the Musée Fabre in Montpellier). The arrangement of the figures in Silvestre's initial sketch strikingly resembles that seen in the frontispiece to the first Discourse: Minerva occupies the position of Prometheus and Prometheus that of the satyr, the figure of Man being placed in the centre.

In the semiotics of paintings and pictures, clouds do indeed tend to indicate hierophany, or the manifestation of the sacred. See Hubert Damish, Théorie du nuage de Giotto à Cézanne: Pour une histoire de la peinture, Paris 1972, p. 67.

Victor I. Stoichita, L'effet Pygmalion. Pour une anthropologie historique des simulacres, Geneva 2008, p. 119-125 of this book are devoted to Prometheus, a figure who is not without connection to that of the sculptor Pygmalion, whose legend was also very dear to Rousseau.

Rousseau, second letter to Malesherbes of 12 th of January 1762: "If anything were ever more like what is called a 'sudden inspiration' [...] ; all of a sudden I felt my mind dazzled by a thousand lights", in: OEuvres completes. Vol. I, p. 1135. Could it be that this a posteriori reading of the Vincennes event is a reading filtered through the recollection also of the allegorical frontispiece of the Discours sur les sciences et les arts?

Barbarus hic ego sum quia non intelligor illis: this quotation from Ovid's Lamentations appears for the first time in Rousseau's work here at the threshold of the first Discourse.

David P. Becker, Drawings for Book Illustration. The Hofer collection, Cambridge, Mass. 1980, p. 24. The author recalls that R. Portalis, at the end of his entry on Gravelot in Les dessinateurs d'illustrations au dixhuitième siècle, mentioned the discovery, in the 19 th century, of this portfolio "in which Gravelot had conserved, one can suppose, almost all the sketches forming the bases of his works. This collection, in which

Described by the brothers Goncourt as "the century of the vignette" (in their chapter on Gravelot in L'Art du dix-huitième siècle, 1868).

Pascal Griener, Gravelot au service de Rousseau et de Voltaire: deux visions opposées de l'illustration, in: Annales Jean-Jacques Rousseau 45 (2003), p. 377-409.

Discours qui a remporté le prix à l'Académie de Dijon, en l'Année 1750. Sur cette question proposée par la même Académie : Si le rétablissement des Sciences & des Arts a contribué à épurer les moeurs. Barbarus hic ego sum, quia non intelligor illis. Ovid., in OEUVRES DE M. ROUSSEAU DE GENEVE. NOUVELLE EDITION. Revue, corrigée, & augmentée de plusieurs morceaux qui n'avoient point encore paru, A Neuchatel, 1764-1768, 9 volumes, Vol. I.

Daniel Ferrer, Variantes et variations: la sonate de Beethoven et le chien de Hogarth, in: Logiques du brouillon. Modèles pour une critique génétique, Paris 2011, p. 130-139.

of the satyr. We note an especial concern with graphically revising the position of the satyr's hand, since this rough draft sketch gives, in fact, the satyr no less than three hands.

These preparatory drawings allow us to observe the draughtsman at work, searching for his motif and placing his figures. Both drawings already bear the signature of the artist: "H. Gravelot inven. 31 ". The original frontispiece's caption is also taken up into this onetruncated, indeed, by the removal of the footnote reference which had helped to specify its meaning in the plate of 1751. But this means that the image can now fully take on the role of a frontispiece, illustrating not a detail but a general vision of the work.

Gravelot's reflection on the image finds its point of departure in his predecessor's artistic proposition, which he cites even as he modifies it at several points, bringing about a shift in its sense and meaning. One discovers here again, indeed, the trio of figures brought together earlier by Pierre: Prometheus, the statue and the satyr. But their positions vis-à-vis one another have changed and, instead of an almost rectilinear procession of the figures in descending hierarchical order against the background of clouds and sky, Gravelot plays on the dimension of depth and creates a spiral movement, the initiating actor in which is no longer Prometheus, but rather the satyr himself, who is now placed in the foreground, one hand extended toward the torch and the other toward the "gazers" (to use a term which approximates, in English, to the French neologism regardants coined by Rousseau), that is to say, toward us, the readers. Prometheus is now placed farther away from the humanitysymbolizing statue and his hand is no longer posed on this latter's shoulder.

Let us now turn to consider the graphic revisions which result, in this sketch, in a doubling of the satyr's left arm. Perhaps misled by the polyvalence of the formulation "you do not know it", in which the pronoun it might possibly also be taken to refer to the still-inert human being embodied in the marble of the statue, Gravelot wavers here and the satyr's hand is drawn by him, initially, as reaching out toward the statue as if to touch it. It is the satyr, in this first draft version, who stands ready to perform the gesture that is no longer Prometheus's: to place his hand upon the inanimate marble, in order to "know it" or, symbolically, to bring it to life. But the sense of touch becomes completely obliterated in the final version of the drawing because, understanding his mistake, Gravelot now raises the satyr's arm so that it is reaching rather toward the torch. Then, in the ink sketch, the hand is drawn arched back even farther so as to make the satyr's intention still more clear. From this point on in the frontispiece's 31 The term abbreviated here is "invenit", since it is the draughtsman who invents the engraving, while the engraver himself only sculpts (sculpsit) it; see supra fig. 1. development, the hand of the satyr will come to occupy the central position within the image, replacing and displacing the fire of Prometheus.

It will maintain this position in the next phase: that of the progressive fabrication of the engraving itself by Joseph de Longueil. De Longueil is yet another "author" of the print, responsible for the intaglio into the metal which gives this latter another form of existence. [START_REF]On the collaborative dimension of creation, see Nicolas Donin/Daniel Ferrer, Auteur(s) et acteurs de la genèse[END_REF] .

To be noted, in this final stage, is the disappearance of the caption which had previously frozen the satyr into the position of the addressee ("you") and had prevented him, as it were, from speaking in the first person.

Through a series of retouches, Gravelot makes of the satyr the true protagonist of the frontispiece, first by moving him into the foreground, then thanks to the play of his hands, one of which now takes the place of the Promethean torch while the other seems to beckon the reader to enter the dance of the image.

But who is this satyr who emerges from the "satyr-drama" of Aeschylus and whom Rousseau, via Plutarch, invites to make his way into the polemical, iconoclastic space of his first Discourse? Who is this satyr who, a minor presence in the text, becomes a major one in the image? In the excellent book that he has devoted to this figure, François Lissarague informs us that, in 6 th -century BC Greecewhen the city-state was just being constructed and established as the ideal form of political co-existence and the main task facing Greece was that of civilizing human violencethe satyr arose as an "open manipulatory model" and an "aberrative factor", [START_REF] Lissarague | La Cité des satyres[END_REF] which made it possible to play with social and cultural codes by contesting these codes through the de-centred, ludic gaze of the goat-man.

The form of the "satyr-drama" represented a way of "playing with culture, which is first, as it were, cast off and then reconstructed through the intermediary of the satyrs, who behave like clueless innocents, discovering for themselves the world which surrounds them along with all its objects." 34 This "play" is surely one and the same as the "play" in which Rousseau himself engages, in his turn and in his own era, by putting radically into question the sciences and the arts while all the while playing with the codes of these latter, so that he too becomes an "aberrative factor" in the society in which he lives. In Gravelot's frontispiece, then, it is perhaps rather the role of the satyr than that of Prometheus that Rousseau himself is called upon to play, particularly as the satyr portrayed here is a satyr who has forfeited something of


This special issue is part of a recent historiographical trend that focuses on the entwined histories of calculation and measurement practices. On the philosophical level, we can mention, for example, the revival of philosophical studies of measurement as witnessed by the Measurement at the Crossroads Conference held in Paris in 2018. 1 In the field of the history of mathematics, this revival has come mainly from scholars who have been interested in the various ways numbers and measurement values have been shaped in the ancient worlds 2 . Finally, the developments of the fledgling field of ethnomathematics have enlightened the diversity of practices of calculation and measurement [START_REF] Vellard | Anthropologie et sciences cognitives: une étude des procédures de calcul mental utilisées par une population analphabète[END_REF][START_REF] Trouré | Une étude ethnomathématique au Burkina Faso : l'arithmétique au quotidien[END_REF]Bednarz 2010, Owens, Lean, Paraide and[START_REF] Owens | History of Number. Evidence from Papua New Guinea and Oceania[END_REF][START_REF] Vandendriessche | Variabilité culturelle de la numératie. Quelques points d'entrée dans la littérature ethnomathématique[END_REF][START_REF] Vandendriessche | Anthropologie des nombres et ethnomathématique[END_REF].

These lines of research have raised historical and historiographical issues on which, in this volume, we would like to bring a particular light. To what extent has the history of numbers and calculation, on the one hand, and the history of metrology and measurement on the other, have been approached separately by historians and philosophers of mathematics? 3 How have ethnographic and anthropological studies of the 19th and 20th centuries impacted analytical works on numbers, up to now? We have chosen to address these questions by focusing on the opposition between "concrete numbers" and "abstract numbers", whatever this opposition may have been conceptualized by ancient authors themselves or theorized by modern commentators.

The present special issue draws on several disciplinary perspectives in order to critically analyze the historiographical opposition between "concrete numbers" and "abstract numbers". Some historians, philosophers, anthropologists or researchers in education have theorized a separation between "numbers" and the entities enumerated or counted with these numbers, and more particularly, between numbers and measurement units attached to them in the expression of measurement values. To what extent does this separation reflect on the practices carried out in societies or social groups under scrutiny by these scholars? How the notion of "abstract numbers"-as opposed to those described as "concrete numbers"-has shaped the history of numerations?

The chapters in this special issue confront common historiography with the great diversity of numeration and measurement systems attested to by the various textual and ethnographic sources available to us. This volume-chapters 3 and 4 in particular-shows how a conceptualization of numbers consisting in isolating/extracting numbers from their context of 1 The Measurement at the Crossroads Conference was part of the events organized around the redefinition of the International System of Measurement (De Courtenay, Gregis, Lacki, and Proust, 2021). It was the third of a conference series devoted to the philosophy of measurement. The previous events were the Dimensions of Measurement Conference, Bielefeld 2013, and The Making of Measurement Conference, Cambridge 2015. 2 The "Science in Ancient World" European Project (ERC-Advanced Grant, 2011[START_REF] Vandendriessche | Variabilité culturelle de la numératie. Quelques points d'entrée dans la littérature ethnomathématique[END_REF], led by Pr. Karine Chemla, was largely devoted to this issue [START_REF] Michel | Mathematics, Administrative and Economical Activities in Ancient World[END_REF]Chemla, Keller, and Proust, 2022). See also [START_REF] Bernard | Le sens des nombres. Mesure, valeur et informations chiffrées: une approche historique[END_REF]. 3 Chemla, 2022 practice, gave rise to a linear history of oral and written numerations rooted in evolutionary theories and classifications.

The chapters devoted to modern arithmetic textbooks (6,7,8,9), beyond the notion of "concrete numbers", address the issue of the treatment, in mathematical practices, of the quantities (in the sense of "magnitude values" according to chapter 6's terminology). How was the problem of multiplying quantities of different dimensions (for example a price by a length) tackled in modern arithmetic textbooks? As rightly pointed out by Ferreira and Schubring (chapter 7), "It is precisely this blind spot that [their] article, and the [present] special issue as a whole, seeks to highlight".

The initial international call for papers for this special issue on "concrete" versus "abstract" numbers was not limited to any period or region of the world. Nevertheless, we (the Editors) received proposals essentially related to three types of historical or historiographical documentation: first, ethnographic observations on numerations; second, historical sources on numbers from the Ancient Near East; and third, elementary arithmetic treatises and textbooks produced between the end of the 18 th century and the beginning of the 20 th century.

What would such focalization mean? Although it's difficult to know, one may observe that the chapters devoted to either the first or second types of documentation focus mainly on historiographical matters, i.e. the way in which some scholars have analyzed the use of numbers and measurement units in ancient or ethnographic sources (chapters 2, 3, 4). In these cases, the opposition between "abstract" and "concrete" numbers reflect the observers' categorizations. Indeed, the ethnographic sources, on the one hand, and the archaeological and epigraphic sources, on the other hand, relating to the birth of writing in various regions of the world, have stimulated many observers' considerations on the alleged emergence of "abstract" numbers, and more generally on the concept of abstraction. On the basis of such considerations, value judgments have been made about the ancient or native actors' capacity for being able (or not) to deal with abstract thought. This paved the way for an evolutionary conception of the history of mathematics, leading to the historical pattern of a unilineal development of numbers from the concrete to the abstract (chapters 2, 3, 4). By contrast, the chapters on modern arithmetic treatises and textbooks (chapter 6, 7, 8, 9) analyze an opposition between "abstract" and "concrete" numbers, or similar dichotomies, that the authors of these arithmetic books explicitly theorize themselves.

French mathematician Etienne Bézout's (1730Bézout's ( -1783) ) definition of "concrete" and "abstract" numbers is certainly the most prevalent and still in use today. This definition is therefore worth considering, but it should be kept in mind that other approaches may have deviated from it (see for example Thureau-Dangin's use of the notion of "abstract numbers" in chapter 4). Bézout defines "abstract" and "concrete" numbers as follows:

A number that is expressed without mentioning the kind of units, as when one simply says three or three times, four or four times, is called an "abstract number"; and when the kind of units is concurrently specified, as when one says four pounds, one hundred barrels, it is called a "concrete number" [START_REF] Bézout | Cours de mathématiques à l'usage des gardes du pavillon et de la marine, Eléments d'arithmétique[END_REF] 1775, 2-3, Section 6, our translation).4

Similar but slightly different categorizations have been introduced before and after him. For instance, in his textbooks on elementary arithmetic, Polish mathematician Stefan Banach relies on a comparable dichotomy that he introduces as an opposition between "denominate numbers" and "non-denominate numbers" (chapter 9). Although absent from this special issue, similar forms of opposition between different classes of numbers can be found in earlier arithmetic treatises and early modern documents.

In histories of arithmetic from the 19 th century onward, the opposition between "abstract" and "concrete" numbers has played a more or less important role, depending on the author. This opposition definitely structures the Arithmetic published by Cambridge algebraist Rev. George Peacock (1791Peacock ( -1858) ) in 1845. Indeed, Peacock divides the historical section (entitled "History of the science") of the paper into Part I Arithmetic of abstract numbers and Part II Concrete numbers. Furthermore, as shown by the historian of mathematics Maris-José Durand Richard ( 2011), Peacock's method is groundbreaking in that he develops a genuine algebraic approach to measurement units.5

In contrast, this opposition is completely absent from A History of Mathematics (1893) by Florian Cajori (1859Cajori ( -1930)), which only deals with so-called pure numbers (that is, "abstract" numbers according to Bezout's definition) and ignores arithmetic with measurement units.

American mathematician David Eugene Smith (1860Smith ( -1944) ) for his part does mention the opposition between concrete and abstract numbers in his book A History of Mathematics (1925). However, he considers it as uninteresting, and actually not useful for mathematicians.

According to him the latter categorization of numbers could be of interest only to elementary mathematics teachers:

ELEMENTARY CLASSIFICATIONS OF NUMBERS

Abstract and Concrete. The distinction between abstract and concrete numbers is modern. The Greek arithmeticians were concerned only with the former, while writers on logistic naturally paid no attention to such fine distinctions. It was not until the two streams of ancient numbers joined to form our modern elementary arithmetic that it was thought worthwhile to make this classification, and then only in elementary school.

The terms "abstract" and "concrete" were slow in establishing themselves. The mathematicians did not need them, and the elementary school teachers did not have enough authority to standardize them. In the 16th century, textbook writers began to make the distinction between pure numbers and numbers to which some denomination was attached, and so we find Trenchant (1566), for example, speaking of absolute and denominate numbers, the latter including not only 3 feet but also 3 fourths. (note 1: quotation excerpted from a 1578 edition). "From that time on the distinction is found with increasing frequency in elementary works. Such refinements, however, as required the product to be of the same denomination as the multiplicand are, in general, 19th century creations of the schools" (Smith 1925, 12).

The importance afforded to the opposition between "abstract" and "concrete" numbers by these different authors is correlated to the way they consider (or do not consider) measurement units as worthy of interest from a mathematical point of view. The controversial role of this opposition is further discussed in chapter 6.

Beyond the opposition between "concrete" and "abstract" numbers, what is the meaning of the concepts of "abstraction" and "concreteness" often put forward in the context of studies devoted to numbers and numerations? What is striking, through the chapters offered in the present collection of papers, is that anthropologists, historians and philosophers of mathematics, as well as mathematicians-at least from the 18 th century onward-use differently the concept of "abstraction"-as well as "concreteness"-when these concepts are applied to the elaboration of numbers. This point is first brought to light in the present special issue by studying how some 19 th until mid-20 th century mathematicians and philosophers, like Levi Leonard Conant (1857Conant ( -1916) ) and Lucien Lévy-Bruhl (1857-1939) in particular, analyzed the numerical systems in use in the so-called primitive societies, documented by 19 th -century ethnographers. These scholars often claimed that these systems would have a more "concrete"-or less "abstract"-character than those developed in more "advanced" societies, due to their close relation to sensible reality (digital counting, use of classifiers, etc.) as well as their mystical aspects specifically (chapters 2, 3). Regarding 19 th and 20 th century historical studies, the "abstract" character of numbers is also put forward in various notions. One of these is Bézout's (chapters 6, 7), but a slightly different conception is used by Thureau-Dangin (chapter 4).

The present special issue consists of a collection of eight chapters divided into three parts.

Part I What is concrete in "concrete numbers"?

This part examines different meanings given to "abstraction" and "concreteness" in the historiography of numbers. How the notion of abstraction, as applied to numbers, has been understood and used in historiography? This historiographical approach is contrasted with evidence from ancient and ethnographic sources.

Eric Vandendriessche (chapter 2) opens the present volume with a chapter aiming at highlighting the scholarly discussions that occurred from the 19th century until mid-20th century on the presumed concreteness of numerations elaborated in the so-called primitiveinferior-barbarous societies. These discussions took place within different disciplinary fields (history of mathematics, philosophy of science, ethnology/anthropology) on the basis of ethnographical sources collected either by missionaries, explorers, or ethnologists, within these societies. Adopting a historiographical approach, and by focusing more particularly on the contributions of some influential 19 th and early 20 th -century scholars-George Peacock (1845), Levi Leonard Conant (1896), Lucien Lévy-Bruhl (1910), Léon Brunschvicq (1912)this chapter highlights how these new sources on (non-written) numbers/numerations have been an opportunity for these scholars to reflect on the supposed abstraction process related to the elaboration of numbers. According to these authors, this process, which would consist in detaching/distancing numbers from what is counted, would have rarely occurred, in their views, with the notable exception of Peacock's more nuanced approach, among "primitive" peoples.

This chapter brings us to the context of colonialism, and the predominant late 19 th century anthropological evolutionist theories, as well as philosophical theories on the "primitive" mentalities (steeped in a profound racism). Indeed, these studies of numerations, as elaborated by "primitive" peoples, have been carried out as part of a more general discussion on the characteristics of the "primitive" mind, and the ability of these people to use abstract and rational concepts. Thus, tracing a red line between humans whose minds are governed by abstraction/rationality and the others. In spite of the criticisms to which they have been subjected, these evolutionary theories are still prevalent today, as the following chapters show.

Miguel Valério and Silvia Ferrara's chapter 3 offers a critical analysis of the historiography of numeracy, in Mesopotamia above all, but also in Egypt, the Aegean world, China and Mesoamerica. They criticize anthropological literature and examples collected by authors such as Conant, who consider that numbers are generally context-dependent. They object to the linear and evolutionist conception of the history of numbers adopted by most historians. In particular, they refute the idea that the history of numbers is that of an evolution from concreteness to abstractness. Rejecting the evolutionist scheme, they differ from Denise Schmandt-Besserat's thesis. Indeed, according to Schmandt-Besserat, the history of numbers in the Ancient Near East goes through three stages:

(1) One-to-one correspondence (2) Concrete counting (3) Abstract counting. Schmandt-Besserat opposes proto-literate (ca. before 3500 BCE) evidence, essentially tokens and signs marked on the external surfaces of bullae or flat pieces of clay, from later written sources on tablets. Tokens and the like would fuse the notions of quality and quantity, while 3 rd millennium written sources would exhibit "abstract" numbers, that is, represented by signs separated from the items quantified, and which has the same form independently of the items quantified. Valério and Ferrara insist that the function of tokens is much more obscure than what is claimed by Schmandt-Besserat because we have very little-or no-information on their archaeological context. Against Schmandt-Besserat, they argue that, "all numbers have an inherent relation to abstraction, even when things are materially counted". As an alternative to the linear evolution from concreteness to abstractness, they claim that the crucial shift in the history of numbers-at least in Mesopotamia-was the introduction of power-based notations. The history of numeracy in Mesopotamia is envisaged in the broader context of the history of writing, which includes two paths, numeracy and literacy as separate (albeit interconnected) phenomena.

Valério and Ferrara conclude by making a connection between the three aspects they analyze in their chapter: 1) the specific history of numeracy, different from that of literacy, 2) the criticism of the dichotomy "abstract" versus "concrete", and 3) the appearance of powerbased systems of numbers.

Christine Proust (chapter 4) shows that the "abstract" character of numbers was understood in quite different ways by historians. In Schmandt-Besserat's scheme, described in chapter 3, the abstract numbers are those which have an autonomous identity, and are represented by signs distinct from those which represent quantified items. In Damerow and Englund's scheme, numbers written in sexagesimal place-value notation are abstract in the sense that they represent a progress in the path toward mathematical abstraction (Nissen, Damerow and Englung 1993). For Thureau-Dangin, numbers written in sexagesimal place-value notation are abstract because of their floating character. Unlike Damerow and Englund, he draws a clear link between the numeration systems and the nature of operations performed with the numbers. Interestingly enough, Thureau-Dangin was the first to introduce the notion of "abstract numbers" in relation to the use of a floating sexagesimal place-value notation, but abandoned this notion later, when he abandoned the idea that these were floating numbers. In his publications, the expression "abstract numbers" disappears together with the introduction of the orders of magnitudes of numbers in translations. This confirms that, for Thureau-Dangin, the "abstract" character of numbers was linked, in his earliest publications, to their "floating" nature, and not to the presence of measurement units after the numbers. Turning to a close examination of cuneiform sources, Proust shows that a diversity of numerical systems coexisted in the same context, or texts, and that these different numerical systems do not reflect different stages of an evolution, but different functions in the same context. Similar observations emanate from documents produced in other parts of the world, for example in India and China (Chemla, Keller, and Proust 2022).

Part II Working on numbers with and without measurement units

This part analyzes the meaning of the presence of measurement units in ancient administrative texts or modern teaching practices. How is the mathematical work on measurement units an integral-and sometimes essential-part of the mathematical elaborations? How does the inclusion of measurement units shape our understanding of numerical systems in past or present treaties and textbooks? How could focusing on measurement units open up new prospects for discussion on mathematical practices and teaching?

The agrarian yield rates, or "šukunnûm" numbers, analyzed in chapter 5 by Baptiste Fiette provide excellent material for a discussion of the notion of "abstraction" conveyed by the expression "abstract number". Numbers representing agrarian yield rates appear, under the name of "šukunnûm", in a corpus of administrative texts from the Larsa region (South Mesopotamia) and dated to the reign of King Hammurabi (ca. 1792-1750 BCE). The particularity of these numbers is that, contrary to the general rule in administrative texts, they are not accompanied by measurement units. What is the meaning of the lack of measurement units after yield rates in the texts considered? In which sense are the šukunnûm numbers "abstract" in this context? Fiette elaborates a complex and nuanced answer to this question. He analyzes in detail the texts containing yield rates, examining their function and use as well as the epigraphy of these numbers. The texts studied (yield tables and harvest registers) were designed by a provincial governor (Šamaš-hazir) to assess the amount of taxes on barley production owed to King Hammurabi. These texts contain data on surfaces, yields and corresponding quantities of grain in tabular format. Other texts such as letters between the governor and the king, a list of crops, and a land sale contract, allow Fiette to specify the function and use of these tables and registers. Evaluating yields requires an estimate of the surfaces and production of these fields. Fiette details the methods used by the governor's surveyors to measure field dimensions, in order to calculate their surfaces. Interestingly, he shows that the adopted metrology of surfaces differs according to the nature and size of the field: the evaluation of large domains providing enormous amounts of barley to the Royal Palace is based on the unit bur (ca. 6.5 ha) and expressed roughly, while that of small individual fields are based on the unit sar (ca. 36 m²) and expressed with high precision values. He suggests that these two approaches reflect different conceptions of surfaces. These conceptions are connected with different uses of estimated yields: to evaluate the taxes to be paid to the king, or "to calculate the agricultural productivity of a team of harvesters in a small area". Various pieces of evidence, such as the content of letters and the dates of the forecast tables, show that the yields have been estimated by "the examination of the first visible shoots of barley […] in early spring". For this reason, the translation of the "šukunnum" number adopted by Fiette is "estimated yield". As we can see, the estimated yields present two facets: they are shaped by incorporating empirical observations of shoots of barley and a rough estimation of the yield (by dividing the surface sample by the estimated production in early spring), and they are used for calculation in advance of the final production (multiply the actual surface by the estimated yield). In conclusion, these tables and registers, in which the yield rates appear, are intended to estimate the expected harvest, and thus to determine the quantities of barley owed to the king. Fiette shows how this context explains the particular form that yield rates take. The author concludes that estimated yields are "abstract" in the sense that they are predictive by nature, and not so abstract in the sense that they result from empirical estimations.

The last contribution (chapter 6) of this second part examines the mathematics education field, highlighting the use-or not-of magnitudes and measurement units in the teaching of arithmetic in French elementary schools, from 1845 until the 2000s, focusing more specifically on the teaching of proportionality. Two major previously introduced sets of conceptual/methodological tools (serving as reading grids) allow the authors, Christine Chambris and Jana Visnovska, to analyze in great detail the various and successive syllabi/curricula that have structured elementary mathematics teaching during this period. The first one is known as the concept of "praxeologies" introduced by a didactician of mathematics Yves Chevallard (1992), in an attempt to study jointly practices (praxis/knowhow) and the discourse (logos) about them. The second set is borrowed from mathematician Hans Freudenthal (1983), who intended to define the different concepts of "quantity", "magnitude", "magnitude value", and "measuring number", while introducing the extremely useful distinction between two types of ratios defined as "internal" (i.e. ratio between values of the same magnitude) and "external" (i.e. ratio between values of different magnitudes).

With this conceptual and methodological framework in mind, Chambris and Visnovska question the "roles of magnitudes and units in the French elementary syllabi" from the mid-19th century onward, and "how these changes were orchestrated", as well as their impact on the "teaching and learning of proportionality". The article brings forward a "long period of relative stability" regarding the latter issue-from the mid-19th century until the 1940swhere the notion of concrete versus abstract numbers was conceptualized in the syllabi as well as in textbooks, while considering magnitudes as a central concept in arithmetic. Nevertheless, it is shown that the teaching of proportionality (and the "rule of three" in particular) has entailed pedagogical debates and disagreements, introducing different approaches, from one period to another, involving the use of either "internal ratios", rates as "external ratios of magnitudes", and rates with "quotient units". The latter approach has thereafter been the most commonly used, until the 1970s, and the setup of the French New Math Reform (1955Reform ( -1975)). Consequently, in addition to the disappearance of the notion of "concrete numbers", the introduction of "linear functions" became the unique conceptual tool for teaching proportionality. As the authors demonstrate, French mathematics education will take a big step backwards, from 2000 onward, reintroducing measurement units, either in formal, or informal calculations. Indeed, all of these pedagogical changes regarding the teaching of basic arithmetic in elementary schools, suggests that the issue of dealing with quantities has not been tackled in a satisfactory way in mathematics in general, in contrast with what has been achieved through theories of measurement in physics. The issue of multiplication and division of quantities (in the sense of "magnitude values") is again addressed in depth in Part III.

Part III Complex and denominate numbers

As an extension of Chambris and Visnovska's analysis, this part delves deeper into the issue of operating with "concrete" numbers. The fundamental question that appears behind the opposition between abstract and concrete numbers, or denominate and non-denominate, or other similar dichotomies, is the distinction between number and quantity of something (or "magnitude values"). Indeed, this distinction plays an essential role in the implementation of multiplications and divisions performed on these entities. In addition, the treatises and textbooks examined in this part highlight the impact of the French Revolution and the Reform of weights and measures on the conceptualizations of numbers. More precisely, it shows how the new decimal system has led to redefining, marginalizing or questioning the notion of "concrete" numbers first in France, then in Europe and Brazil.

Débora Ferreira and Gert Schubring (chapter 7) focus on a specific sort of concrete number, the "complex numbers" as defined by Camus (1749), Bézout and other French authors quoted in chapter 8. According to Bézout, among the concrete numbers, some are "complex" (i.e. composed of several juxtaposed elements, such as "2 hours and 20 minutes"), others are "uncomplex" (i.e. composed of a single element, such as "2 hours"). The importance of this distinction for Bézout appears through his reflections on the commutativity of the multiplication of concrete numbers. He shows that the multiplication of a complex number by a concrete number does not provide the same results as the inverse multiplication. Ferreira and Schubring critically discuss Bézout's approach of the commutativity of the multiplication and detect a "trick" in Bézout's reasoning. In order to solve the problems posed by Bézout, they open the broader issue of the multiplication of quantities, hence the dimension of the product. Ferreira and Schubring thus analyze in depth the links between numbers, quantities and operations in modern arithmetic textbooks.

Bézout exhibits an example showing that the product of two concrete numbers, one representing a length and the other a price, is different if the factors are switched. How does Bézout explain this phenomenon? His explanation relies mainly on the fact that for him, the product must be of the same nature as the multiplicand. Ferreira and Schubring highlight that Bézout's argument is directly related to his definition of multiplication. Indeed, multiplication is defined "canonically" as an iterated addition, in Euclid's style. In this dissymmetric definition, one of the factors acts as a multiplicand (the quantity to be iterated), the other as a multiplicator (the number of times the quantity is added to itself), which implies that the dimension of the product is that of the multiplicand. However, this definition concerns only the product of two numbers (integers, fractions and even irrationals by extension), and the product of a quantity of something by a number. It does not work for the product of two quantities of something, which raises the problem of the result's dimension.

Yet, according to Ferreira and Schubring, in his examples, Bézout multiplies two quantities of different dimensions (a length with a price). When he claims that the product is of the same nature as the multiplicand, he does not question his definition of the multiplication and the dimension of the product. In his Géométrie (1770, p. 89), Bézout refuses to multiply a line by a line, but, according to Ferreira and Schubring, he tries to multiply geometrical quantities by a method which "revives the notion of indivisibility". The question of the multiplication of geometric quantities had been addressed before Bézout, notably by Camus, who adopts another point of view. For Camus, the units of the product are products of the units: toisetoise, toise-foot, etc. "Camus was well aware that the dimension of the product has to be different from those of the factors". But, unlike Bézout, Camus does not consider the problem of commutativity. The authors of arithmetic textbooks after Bézout, in France as well as in other regions of Europa and in Brazil, point out the fact that, in the multiplication of complex numbers, one cannot switch the factors. Two reasons are generally given to explain this noncommutativity: the first and most frequent one is, according to Bézout, that the dimension ("specie") of the product is that of the multiplicand. The second one (minor except in a few cases as in Ottoni's arithmetic textbooks where it occupies the first place) is that, if the metrologies of the two factors are different, the rules of calculation are different. Only in the 20 th century, some physicists and mathematicians, notably Hans Freudenthal , did "solve the riddle" by considering quantities as "tensors", thus regarding quantities as "proper domains of mathematical entities".

Xiaofei Wang's chapter 8 concerns the context of the establishment of the decimal system reform initiated during the French Revolution (1789-1799) and onward during the First Empire (1804-1814), by mathematicians/astronomers of the time, at the École Normale de l'an III in particular, which thereafter became the École Polytechnique. The chapter begins by highlighting the fundamental role played by mathematician Joseph-Louis Lagrange (1736-1813), at the turn of the 19th century, in the promotion and dissemination of the decimal arithmetic that allows-in Lagrange's view-"regularity and uniformity to the operations of arithmetic." As Wang brings to light, Lagrange conceived of Arithmetic as the "science of calculation" (in contrast to a "science of numbers"). This approach led him to reflect on and compare different calculation methods in the history of mathematics-methods obviously linked to the different systems of numeration. Consequently, he notably contributed to a renewed interest in the arithmetic of the Ancient Greeks.

Another important scholar of the time, the astronomer and mathematician Jean-Baptiste Delambre (1749-1822), shared with Lagrange the latter's interest in ancient arithmetic. As shown by Wang, his contribution on the topic is of fundamental importance, affording an original approach (and reconstruction) to calculation as it was presumably carried out by the Ancient Greeks. To achieve this, the chapter takes a detour by the description of the so-called complex numbers, defined as "concrete numbers composed of several parts related respectively to different [measurement] units". Commonly used before the French Revolution (see chapter 7), these complex numbers (and the sophisticated calculation methods that their use entails) have coexisted with the weights and measurements decimal system, particularly in textbooks, over several decades. In a very original way, Delambre used the calculation on complex numbers (cf. Bezout), as a modeling tool to explore the arithmetic of the Ancient Greeks. He argued that Greek numbers can be rewritten/translated into numbers in a form similar to "complex numbers", based on regular (and decimal) subdivisions of numbers. According to him, such rewriting of Greek numbers allows us to hypothesize on the arithmetical methods carried out in Ancient Greece.

The last chapter of this volume offered by historian of mathematics Karolina Karpińska, highlights a little-known part of the famous Polish mathematician Stefan Banach's (1892Banach's ( -1945) ) professional life. In the interwar period, Banach-facing some financial difficultieswrote several textbooks (as author or co-author) for secondary schools. This occurred in the context of the reform of the curriculum that followed Poland's independence (from Prussia, Austria, and Russia, in 1918). Some of these publications are devoted to the teaching of arithmetic, and became some of the most popular mathematics textbooks used in Polish schools during this period. Karpińska analyzes the latter publications in comparison to other textbooks published at the same period, as well as to the textbooks that Banach used as a student of the "4th Gymnasium" in Kraków. This affords us a better understanding of Banach's influence, as well as his original contributions to the field of elementary mathematics education. In addition to the introduction of the different kinds of numbers (integer, fractional, relative, rational, irrational), Banach presents the concept of liczby mianowan-literally "denominate numbers"-aiming at expressing measures of lengths, weights, volumes, and other "magnitude values" to use chapter 6's terminology. He considers these denominate numbers as singular numerical expressions, elaborated from numbers (sometimes also defined as liczba niemianowana-literally "non-denominate numbers"). In contrast to other Polish authors of the time, Banach's attempt is to make his readers perceive the very nature of these numerical expressions through the operations that can be performed on (simple or compound) denominate numbers. As shown by Karpińska, the latter numerical expressions are indeed of fundamental importance to Banach's approach of numeracy, in its links to other disciplines like physics, geography, among others.

French original text: « Un nombre qu'on énonce sans désigner l'espèce des unités, comme quand on dit simplement trois ou trois fois, quatre ou quatre fois, s'appelle un nombre abstrait ; et lorsqu'on énonce en même temps l'espèce des unités, comme quand on dit quatre livres, cent tonneaux, on l'appelle nombre concret. »

Regarding Peacock's contribution see also chapter 2 of the present volume.