Paul-Emile Maingé

Eric Kamwa
email: eric.kamwa@univ-antilles.fr

Gilles Joseph
email: gilles.joseph@univ-antilles.fr

Paul-Emile Mainge
email: paul-emile.mainge@univ-antilles.fr

How to obtain an overall reduction of individual income deprivations by means of a finite sequence of T 2 -transformations? A solution and a well-suited algorithm

Keywords: Inequality, Income, Deprivation, T 2 transformation, Algorithm

come

How to obtain an overall reduction of individual income

deprivations by means of a finite sequence of T 2 -transformations? A solution and a well-suited algorithm

Introduction

Income inequality refers to the extent to which incomes are distributed unequally among a population. 1 Progressive transfers (see [START_REF] Dalton | The Measurement of Inequality of Incomes[END_REF]) are the most popular mechanisms often used for the reduction of income inequality. A progressive transfer induces an income transfer from richer to poorer individuals. Nonetheless, while helping in reducing income inequality between two people, it is known that one of the main drawbacks of progressive transfers is that they can increase the income differences between each of these two people and the others. This led some authors to reject the principle of progressive transfers in a search of inequality reduction (see for instance, Amiel andCowell (1992, 1999). Income differences can also allow to define the individual's deprivation as people compare themselves with some reference individual or group of individual within the society rather than with the whole society. See for instance, the papers by [START_REF] Chakravarty | Income inequality and relative deprivation[END_REF], [START_REF] Kakwani | The relative deprivation curve and its applications[END_REF], [START_REF] Runciman | Relative Deprivation and Social Justice[END_REF], [START_REF] Silber | Handbook of Income Inequality Measurement[END_REF] and [START_REF] Yitzhaki | Relative Deprivation and the Gini Coefficient[END_REF][START_REF] Yitzhaki | On an extension of the Gini inequality index[END_REF].

According to [START_REF] Chateauneuf | Does the Lorenz curve really measure inequality? GRAPE Discussion Paper[END_REF] 2 , a way to reduce income inequality while acting on individual deprivations is to introduce a notion of solidarity among the individuals concerned by the transfer process. They suggested some mechanisms (transformations) of which progressive transfers are particular cases. The successive applications of these transformations will result in a distributional improvement according to the individual deprivations. Among the suggested transformations, we are concerned with the T 2 -transformations which stipulates that if some income is taken from a rich individual, then the same amount has to be taken from every non poorer individual and it is no longer necessary that individuals poorer than the transfer recipient benefit also from some equal additional income. When a less unequal income distribution x can be obtained from an income distribution y by means of a finite sequence of T 2 -transformations, no individual will experiment an increase of his total deprivation, some individuals may notice a reduction while others may record no change. Is it not possible to do better? To do better would be to (always) ensure a situation in which by means of T 2 -transformations, each individual experiments a reduction in his total deprivation. That is our objective in this paper. So, we define the conditions for which a finite sequence of a T 2 -transformation will guarantee that every individual experiments a reduction in his total deprivation and we provide an algorithm for such a possibility.

The rest of the paper is organized as follows. In Section 2, we set the framework with some basic notations then we formally define the problem under consideration in this paper. Our main results are provided in Section 3 where we also suggest a well-suited algorithm for solving our problem. Section 4 concludes.

Basic definitions and the considered problem

Basic definitions

Let U denotes the set of income distributions z in IR andd i=1 z i = S (for some fixed positive value S ). In the distribution z, the incomes are sorted by ascending order; so z 1 designates the income of the poorest individual and z d the income of the richest individual.

d such that z = (z 1 , z 2 , . . . , z d-1 , z d ) in IR d (with d ∈ IN * ), 0 ≤ z i ≤ z j for any 1 ≤ i ≤ j ≤ d
We introduce the linear mapping D :

IR d → IR d-1 defined for z in IR d by D(z) = (D 1 (z), D 2 (z), ..., D d-1 (z))
where components are given for i = 1, .., d -1 by

D i (z) = (d -i)z i - d j=i+1 z j
The vector D(z) is the distribution of the individual total income deprivations D i (z) which is, for each individual i, the sum of income pairwise differences between z i and the z j such that z j > z i .

Given two income distribution x and y in U n , x is obtained from y by means of a progressive transfer, if there exists a positive value ∆ (∆ > 0) and two individual i and j such that • x g = y g for all g i, j,

• x i = y i + ∆ and x j = y j -∆ with ∆ ≤ y j -y i 2 .
In principle, a progressive transfer does not reverse the relative positions of the individuals (the donor(s)) who contribute to the transfer and those (the recipient(s)) who benefit from it. As in [START_REF] Chateauneuf | Does the Lorenz curve really measure inequality? GRAPE Discussion Paper[END_REF], we assume that the progressive transfer is rank-preserving in the sense that the relative positions of all the individuals are unaffected; this is tantamount to impose the following condition to a progressive transfer:

∀p q, (x q -x p )(y q -y p ) ≥ 0 (1) A vector x in IR d is obtained from y in IR d by means of a T 2 -transformation, denoted by x = Γ h,k (
y), if there exist a positive value and two indexes h ∈ {1, .., d -1} and k ∈ {1, .., d} with k ≥ h + 1 such that Equation (1) holds and

• x i = y i for i ∈ {1, .., h -1} ∪ {h + 1, .., k -1}, • x h = y h + ∆ with ∆ = (d -k + 1) , • x i = y i -for i = k, .., d,
Let us use an example to illustrate how a T 2 -transformation operates. This example will also motivate the concern of this paper. u = (3,4,6,6,6) = Γ 1 3,5 (y) v = (3,5,5,6,6) = Γ 1 2,5 (y) w = (4,4,5,6,6) = Γ 1 1,5 (y) By computing the corresponding vectors D(.), we get:

Distributions D(.) y (-10, -6, -3, -1) u (-10, -6, 0, 0) v (-10, -2, -2, 0) w (-5, -5, -2, 0)
It is known that when a distribution x is obtained from a distribution y by means of a T 2 -transformation, x is always less unequal than y. More, while going from y to x, the guarantee is that no individual will experiment an increase of his total deprivation. In some cases, some individuals will not experience any change in their total deprivation while others will notice a reduction. What we get in Example 1 gives an illustration of that. In this paper, we want to focus on how to reach a situation in which each individual records a reduction in his total deprivation while obtaining a distribution x from a distribution y by means of T 2 -transformations. This defines the main problem we deal with. Let us formally define our problem.

The considered problem

Given two vectors x and y in U, we wish to show that x can be obtained from y by means of a finite sequence

(z n ) of T 2 -transformations in U such that D(x -y) = D(x) -D(y) > 0,
in the sense that:

• z 0 = y, • (z n ) ⊂ U,
• z n+1 is obtained from z n by a T 2 -transformation,

• (z n ) → x finitely.
Let (z n ) be a sequence in U such that z 0 = y and z n+1 = Γ n h n ,k n (z n ) where h n and k n are indexes such that h n ∈ {1, .., d-1}, k n ∈ {1, .., d} and k n ≥ h n + 1 for all n ≥ 0. For the sake of simplicity, we set D n = D(δ n ) with δ n = xz n .

Main results

As we going to see, the solution to our problem is to establish sufficient conditions so that the goal is achieved.

Sufficient conditions

Propositions 1 and 2 describe how to set the parameters of the transformation in order to ensure that z n+1 = Γ n h n ,k n (z n ) belongs to U. These propositions define the sufficient conditions for z n ∈ U and D(z n ) ≤ D(z n+1 .

Proposition 1. For any n ≥ 0, we have D(z n+1 ) ≥ D(z n ).

Proposition 2. Suppose that z n ∈ U for some n ≥ 0. Then the next iterate z n+1 belongs to U provided that each of the following conditions are satisfied:

n ≤ z n h n +1 -z n h n d -k n + 1 (if k n > h n + 1), (2)
n ≤ z n k n -z n k n -1 (if k n > h n + 1), ( 3
) n ≤ z n h n +1 -z n h n d -h n + 1 (if k n = h n + 1). (4) Proof. It is easy to see that d i=1 z n+1 i = d i=1 z n i = S , so it remains to ensure that z n+1 i ≤ z n+1 j for 1 ≤ i ≤ j ≤ d.
Clearly by the definition of the T 2 -transformation, it suffices to ensure that z n+1 h n ≤ z n+1 h n +1 and z n+1 k n -1 ≤ z n+1 k n . We recall that

z n+1 h n = z n h n + ∆ n , where ∆ n = (d -k n + 1) n z n+1 k n = z n k n -n .
Suppose that k n > h n +1. Then we have z n+1 h n +1 = z n h n +1 , so that the condition

z n+1 h n ≤ z n+1 h n +1 is equivalent to z n h n +∆ n ≤ z n h n +1 , namely (2). Moreover, we have z n+1 k n -1 = z n k n -1 , so that the condition z n+1 k n -1 ≤ z n+1 k n becomes z n k n -1 ≤ z n k n -n , that is (3).
In the case when k n = h n + 1 we have z n+1

h n +1 = z n h n +1n , so that the condition z n+1 h n ≤ z n+1 h n +1 (which also reduces to

z n+1 k n -1 ≤ z n+1 k n ) becomes z n h n + ∆ n ≤ z n h n +1
n , that is (4). First, let us highlight some important relationships that will be useful in the sequel.

Remark 1. Let us set G n = D(z n+1 -z n ). It is easily checked that • G n i = 0 for i = 1, ..., h n -1 • G n h n = (d -h n + 1)∆ n • G n i = ∆ n for i = h n + 1, ..., k n -1 (if k n > h n + 1) • G n i = 0 for i = k n , .
.., d -1. Proposition 3 provides the sufficient conditions for D(z n ) ≤ D(x).

Proposition 3. Suppose that D n ≥ 0 and the following conditions are reached:

• (d -h n + 1)∆ n ≤ D n h n , or equivalently n ≤ D n h n (d -h n + 1)(d -k n + 1) , • ∆ n ≤ D n i for i = h n + 1, ..., k n -1 (if k n > h n + 1), or equivalently n ≤ D n i (d -k n + 1) ,
Then it holds that D(z n+1 ) ≤ D(x).

Proof. Clearly, by linearity of D(.), we have

D(x -z n+1 ) = D(x -z n ) -D(z n+1 -z n ), that is D(x -z n+1 ) = D n -G n . As a consequence, D(x -z n+1 ) ≥ 0 is equivalent to G n ≤ D n
, which obviously leads to the desired result.

As one can notice, the relations defined in Propositions 2 and 3 essentially concern how the choice of parameters should be made; a choice that is crucial given the objective pursued. Equipped with these relation, we are now in position to construct an algorithm for solving our problem in concern here.

Construction of a well-suited algorithm

Prior to propose an algorithm for solving our problem, we first need to state some results that open the way to the implementation of this algorithm. Propositions 4 to 6 describe preliminaries for Propositions 7 to 8 which assures us the solution to our problem, that by a T 2 -transformation, each individual benefits from a reduction of his total deprivation. Proposition 4. For i = 1, ..., d -2, the following estimates are reached:

e1) D n i+1 -D n i = (d -i)(δ n i+1 -δ n i ); e2) D n i+1 -D n i = (d -i)(x n i+1 -x n i ) + (d -i)(z n i -z n i+1 ); e3) D n i+1 -D n i ≥ (d -i)(z n i -z n i+1 ); Proof.
Using the definition of D n i , we have

D n i+1 -D n i = (d -i -1)δ n i+1 - d j=i+2 δ n j -(d -i)δ n i + d j=i+1 δ n j = (d -i -1)δ n i+1 -(d -i)δ n i + δ n i+1 = (d -i)(δ n i+1 -δ n i ),
that is (e1). From the definition of δ n i , we also have

δ n i+1 -δ n i = (x n i+1 -z n i+1 ) -(x n i -z n i ) = (x n i+1 -x n i ) + (z n i -z n i+1 ),
which by (e1) leads to (e2). Moreover observing that x n i+1x n i ≥ 0 (as x ∈ U), we get

δ n i+1 -δ n i ≥ z n i -z n i+1 , (5)
which by (e1) leads to (e3).

Proposition 5.

For i = 1, ..., d -1, D n i ≤ (d -i)(z n d -z n i ) -(d -i)(x i+1 -x i ).
Proof. We recall that

D n i = (d -i)(x i -z n i ) -d j=i+1 (x j -z n j )
. Then, by x j ≤ x j+1 (hence x j ≥ x i+1 for j ≥ i + 1), we obtain

D n i ≤ (d -i)(x i -z n i ) -(d -i)x i+1 + d j=i+1 z n j .
Hence, from z n j ≤ z n j+1 (hence z n j ≥ z n d for j ≥ 0), we get

D n i ≤ (d -i)(x i -z n i ) -(d -i)x i+1 + (d -i)z n d ,
or equivalently

D n i ≤ (d -i)(z n d -z n i ) -(d -i)(x i+1 -x i )
, that is the desired result. Proposition 6. The following properties are satisfied: p2) is obtained from Proposition 5 by the fact that x i+1x i ≥ 0 together with D n i > 0

p1) For i = 1, .., d -2, z n i+1 = z n i =⇒ D n i+1 ≥ D n i ; p2) For i = 1, .., d -1, D n i > 0 =⇒ z n i < z n d . Proof. (p1) is obviously deduced from Proposition 4. (
Proposition 7. Suppose that D n > 0 and set j n = max{i ∈ {1, .., d-1} | D n i > 0}. Then it holds that D n j n > 0 and z n j n +1 > z n j n . Proof. Clearly, we have either z n j n +1 > z n j n or z n j n +1 = z n j n . The latter situation in light of Proposition 6 ensures that

D n j n +1 ≥ D n j n (hence D n j n +1 > 0) when j n ≤ d -2, which is absurd. It is also observed from Proposition 6 that z n d > z n d-1
whenever D n d-1 > 0. This leads immediately to the desired result. Remark 2. Recall that G n = D(z n+1z n ). It is then immediate that G n = D(xz n ) -D(xz n+1 ), or equivalently D n+1 -D n = -G n . This observation together with Remark 1 amounts to

• D n+1 i = D n i for i = 1, .., h n -1 • D n+1 h n = D n h n -(d -h n + 1)∆ n = D n h n -(d -h n + 1)(d -k n + 1) n • D n+1 i = D n i -∆ n for i = h n + 1, ..., k n -1 (if k n > h n + 1) • D n+1 i = D n i for i = k n , ..., d -1.
Proposition 8. Set α n j = z n j+1z n j and β n j = D n j dj . Then the following results hold:

(r1) β n d-1 ≤ α n d-1 ; (r2) For j = 1, ..., d -2, α n j < β n j =⇒ D n j+1 > 0;
Proof. Let us prove (r1). Clearly, we have

α n d-1 = z n d -z n d-1
, while it is a simple matter to see that

β n d-1 = D n d-1 = (x d-1 -z n d-1 ) - d i=d (x i -z n i ) = (x d-1 -z n d-1 ) -(x d -z n d ) = (z n d -z n d-1 ) -(x d -x d-1 ).
As a consequence, we have

β n d-1 = α n d-1 -(x d -x d-1
), so that β n d-1 ≤ α n d-1 (due to x d ≥ x d-1 as x ∈ U). Now we prove (r2). Suppose that α n j < β n j for some j = 1, ..., d -2, hence, we equivalently have

D n j -(d -j)(z n j+1 -z n j ) > 0. (6)
Moreover, using the definition of D n j we have

D n j -(d -j)(z n j+1 -z n j ) = (d -j)(x j -z n j ) -d i= j+1 (x i -z n i ) -(d -j)(z n j+1 -z n j ) = (d -j)(x j -z n j+1 ) -d i= j+1 (x i -z n i ) = (d -j)(x j -x j+1 ) + (d -j)(x j+1 -z n j+1 ) -(x j+1 -z n j+1 ) -d i= j+2 (x i -z n i ) = (d -j)(x j -x j+1 ) + (d -j -1)(x j+1 -z n j+1 ) -d i= j+2 (x i -z n i ) It follows that D n j -(d -j)(z n j+1 -z n j ) = (d -j)(x j -x j+1 ) + D n j+1 .
Consequently, inequality (6) can be rewritten as D n j+1 > (dj)(x j+1x j ); hence, recalling that x j+1x j ≥ 0 (as x ∈ U), we deduce that D n j+1 > 0, that is the desired result.

Remark 3. Suppose for some n ≥ 0 that z n+1 = Γ n h n ,k n (z n ) with k n = h n + 1 (for some n ≥ 0 and h n ∈ {1, .., d -1}), where z n satisfied z n ∈ U and D(y) ≤ D(z n ) ≤ D(x).

According to Propositions 1, 2 and 3, we then have

z n+1 ∈ U and D(y) ≤ D(z n+1 ) ≤ D(x), (8)
provided that n ≤ min{

β n h n d -h n + 1 , α n h n d -h h + 1 },
where α n j = z n j+1z n j and β n j = D n j dj .

We can now introduce an algorithm for solving our problem defined in Section 2.

Algorithm 1 The considered Algorithm Initialization: Set z 0 = y and n = 0;

Step 1: set h n as the largest value i ∈ {1, .., d -1} such that D i (xz n ) > 0;

Step 2: set k n = h n + 1 ;

Step 3:

set n = D n hn (d-h n )(d-h n +1) ; Step 4: Compute z n+1 = Γ n h n ,k n (z n ) (which ensures that D h n (x -z n+1 ) = 0);
Step 5: If z n+1 x then n ← n + 1 and go to Step 1; otherwise stop.

Theorem 1. The sequence (z n ), generated by the considered Algorithm 1, is included in U and converges finitely to x.

Proof. Recall that condition ( 7) is satisfied at the first iteration n = 0. To get the result, we prove that (at each iteration n) condition (7) implies that (8) holds and we face one of the following situation:

(A n ) z n = x; (B n ) h n+1 ≤ h n -1.
To that end we consider the values α n j = z n j+1z n j and β n j = D n j dj introduced in Prop. 8 and Remark 3. Suppose that z n x and the condition (7) holds at some iteration n. It is clear that h n is well-defined and we additionally have x h n +1 > x h n and D n h n > 0 (thanks to Prop. 7). We also have β n h n ≤ α n h n (from Prop. 8), so that the choice of n in the algorithm satisfies n = min{ ≤ D n j for j = 1, ..., d -1 (from Remarks 2 and 3), so that h n+1 ≤ h n -1.

Conclusion

In this paper, we have defined the conditions for which a finite sequence of T 2 -transformations will always guarantee that when moving to a less unequal income distribution, every individual experiments a reduction in his total deprivation and we have provided an algorithm for such a possibility.

Example 1 .

 1 Assume d = 5 and consider the following income distribution: y = (3, 4, 5, 6, 7). The reader can easily check what follows:

  n +1 }. It follows that (8) holds (from Remark 3) and we immediately deduce that D n+1 h n = 0. It is then easily checked that D n+1 j = 0 for j = h n , ..., d -1, because 0 ≤ D n+1 j

Acknowledgements. The authors are grateful to Florian Labarre for checking all the proofs.


The Denishawn school of dance was founded by Ted Shawn and Ruth St Denis in 1915. The New Dance Group, founded in New York City in the early 1930s, was a group of leftwing artists (dancers

Dance in Walt Whitman's Leaves of Grass: Haptic Connectedness and Lyric Choreography Adeline Chevrier-Bosseau, Université Clermont Auvergne

When reading what the pioneers of American dance wrote about their vision and plans for the renewal and rejuvenation of the art form, it is striking to see how omnipresent Walt Whitman, the self-professed "poet of the body" (LG, p. 43), is. Isadora Duncan, one of the greatest reformers of dance in the twentieth century, famously listed Walt Whitman as one of her "dance masters," along with Rousseau and Nietzsche, and defined herself as his "spiritual daughter." 1 His poetry appears as a symbol of Americanness and a template for bold formal innovation in Ted Shawn's manifesto The American Ballet, published in 1926, as well as in Lincoln Kirstein's Dance, A Short History of Theatrical Dancing, published in 1970. 2 Whitman's celebration of the organic truth of the body found a particularly vibrant echo in the Denishawn school of dance, and in the works of all the founders of American modern dance, from the New Dance Group to Martha Graham, for whom the body was the ultimate locus of individual and collective truth. 3 In "Whitman and Modern Dance," Joann Krieg delineates how Whitman's vision of the body, his views on questions of class and race, and his conception of American democracy resonated for the choreographers of the 1930s, particularly Helen Tamiris (who created her Walt Whitman Suite, inspired by Leaves of Grass, in 1934). 4 While critics have investigated the impact of Walt Whitman's poetry on dancers, dance reformers and choreographers, the role and place of dance in Walt Whitman's Leaves of Grass remains widely overlooked. Yet it is one of the performing arts, along with opera and the theatre, which contributed to shaping Whitman's conception and performance of the lyric self. This paper examines how dance factors into Whitman's representation of the body and of the lyric self as an embodied, physical-sometimes athletic-being. I will begin by relating Whitman's evocation of dance in Leaves of Grass to the cultural and social significance of dance in mid-nineteenth-century America, where it functioned as a telltale sign of hierarchies of class. I will then turn to a close reading of "The Sleepers" (LG, p. 356-364), postulating that, in this poem, Whitman develops a unique form of lyric choreography, where the construction of the self as well as the lyric relation between the self and the addressee(s) or other selves are materialized and embodied as a dance.

I. Dance, class and the body in Leaves of Grass

Bard or faun sounding his "barbaric yawp over the roofs of the world" (LG, p. 77), Whitman's lyric self roams through Leaves of Grass, brushing against a myriad bodies and engaging in various Dionysian dances. There are several moments in Leaves of Grass when the speaker explicitly represents himself dancing with other dancers in Dionysian enjoyment, as in "Native Moments" where he exclaims: "I am for those who believe in loose delights, I share the midnight orgies of young men, / I dance with the dancers and drink with the drinkers" (LG, p. 94). 5 In these scenes, dance-usually coupled with drinking and laughter in classic Dionysian fashion-is the physical expression of the exultation of the body. When social dance is depicted, it is always free and vigorous, as in the 15 th section of "Song of Myself": "The bugle calls in the ball-room, the gentlemen run for their partners, the dancers bow to each other" (LG, p. 37). 6 It is always represented as a popular form of entertainment, devoid of any rigid social etiquette, testifying to Whitman's affinity to the working class.

When Whitman wrote Leaves of Grass, social dancing was highly codified in the middle and upper classes. Between 1840 and 1860, as Ann Wagner notes, American culture became increasingly divided between highbrow and lowbrow, which greatly impacted dance as an art form and a social activity. 7 Among the elite assemblies, dancing was to be respectful, elegant, and mindful of the proper etiquette as defined in the many dance manuals that kept rolling out of the printing presses at that time. In the rowdier popular balls, on the other hand, no etiquette was necessary. 8 The first kind was above all social, and gaining entry to these balls was instrumental in someone's upward social mobility, whereas the second kind was merely entertainment for its own sake. 9 Wagner further explains that only high society balls and assemblies, as well as attendance to "respectable" theatrical venues, were regulated by a set of etiquette rules-working-class amusements being wholly dismissed as improper and rude. 10 The working-class danced mainly in informal settings such as barns, saloons, or outside, thus operating in the periphery of prescribed genteel amusements. While physical interactions were highly codified in upper-class assemblies, no such rules applied to the working-class, so that people were free to interact more spontaneously: if touching bare skin, using coarse language or dancing vigorously, holding one's partner firmly, were strictly prohibited in the upper-class, it was generally accepted in working-class balls. 11 Studies by Ann Wagner, Kathy Peiss or Lawrence Levine have shown how such popular amusements were instrumental in creating American popular culture and enforcing a sense of belonging to a particular social class; attending venues in popular, lowbrow places like the Bowery theatre 5 Section 3 of "Song of Myself" too represents the self's "satisfaction" through dance, laughter and song: "I am satisfied-I see, dance, laugh, sing" (LG, p. 28). Dance is also one of the "joys" evoked in "Song of Joys": "Joy of sweet music, joy of the lighted ball-room and the dancers"; "O to have life henceforth a poem of new joys!/ To dance, clap hands, exult, shout, skip, leap, roll on, float on!" (LG, p. 154, 155). 6 Other examples include the "bull-dances" of section 33 of "Song of Myself" (LG, p. 55), the "young men" dancing in "Our Old Feuillage ("Some of the younger men dance to the sound of the banjo or fiddle" [LG, p. 147]), social dancing in the city in section 4 of "Song of the Broad-Axe" ("They fill their hour, the dancers dance, the musicians play for them", [LG, p. 159]), or the "wedding-dances" of section 12 of "Song of the Open Road" (LG, p. 132). 7 Ann Louise Wagner, Adversaries of Dance: From the Puritans to the Present, Urbana and Chicago, University of Illinois Press, 1997. See also Lawrence Levine, Highbrow/Lowbrow, The Emergence of Cultural Hierarchy in America, Cambridge, MA, Harvard University Press, 1988. 8 See A. L. Wagner, Adversaries of Dance, op. cit., p. 127-136. In addition to these advice manuals there were dance manuals, such as Charles Durang's Terpsichore, teaching American readers the new European dances, like the waltz or the polka, which reached America in the late 1840s (ibid., p. 144-145). 9 Ibid, p. 107. or dancing outside the realm of genteel balls-as Whitman did-was thus a strong marker of class identity. 12 Only among the working classes, then, was dance a vigorous and free affair as depicted in Whitman's poems. In the 7 th section of "Song of the Exposition," upper-class dancing is accordingly portrayed as crepuscular, "unhealthy" entertainment:

Away with old romance! Away with novels, plots and plays of foreign courts, Away with love-verses sugar'd in rhyme, the intrigues, amours of idlers, Fitted for only banquets of the night where dancers to late music slide, The unhealthy pleasures, extravagant dissipations of the few, With perfumes, heat and wine, beneath the dazzling chandeliers. (LG, p. 170)

Where working-class dance scenes always evoke a rowdy assembly of young and healthy dancers, this dance scene is associated to the outdated, aristocratic Old World-and therefore unfit to Whitman's democratic America-, as is visible with the reference to "foreign courts" or "banquets." These aristocratic dancers are referred to as "idlers," as opposed to the workers dancing to the banjo in "Our Old Feuillage" for example, and the number of dancers is limited ("the unhealthy pleasures, extravagant dissipations of the few", my emphasis), contrary to working-class dance scenes. Whitman's use of the adjective "unhealthy" to refer to this form of upper-class entertainment is evocative in the context of nineteenth-century discourses on dance and health. Dance adversaries often deemed ballroom dancing "unhealthy" and harmful for the health of young ladies in particular, and identified the heat and late hours-as Whitman does in this section of "Song of the Exposition" -as the chief culprits of ill-health, along with promiscuity 13 . Dancing in balls, they thought, was harmful because it kept people up late, and the heat and proximity of all these bodies in a confined space was bound to foster the spread of contagious diseases. 14 On the contrary, the spontaneous, unrestrained dancing of the lower classes-outdoors (as in "Our Old Feuillage" or "We Two Boys Together Clinging" for example) or in less stuffy ballrooms -was never viewed as unhealthy.

As many critics, like Jimmie Killinsgworth, Harold Aspiz, Michael Moon, or Tenney Nathanson, have noted, Whitman's interest in the body, the physical force and beauty of the human body, its kinetic and erotic expression, permeates Leaves of Grass. 15 Killingsworth reminds us that Whitman reviewed Dr Edward Dixon's book Woman and Her Diseases-a plea for physical exercise as a remedy for many of the ailments afflicting the upper-classes 12 See note 7 above. See also Kathy Peiss, Cheap Amusements: Working Women and Leisure in Turn-of-the-Century New York, Philadephia, Temple University Press, 1986, p. 100-104. 13 This is also congruent with Whitman's interest in health issues and health policies. In his newly recovered series entitled "Manly Health and Training", published in the New York Atlas, Whitman urges sedentary men, students or clerks, to go out and exercise outdoors, for being confined indoors for hours is detrimental to their health. To the student he writes: "Let nothing divert you from your duty to your body. Up in the morning early! Habituate yourself to the brisk walk in the fresh air-to the exercise of pulling the oar-and to the loud declamation upon the hills, or along the shore." And he similarly encourages the clerk to exercise ("To you, clerk, literary man, sedentary person, man of fortune, idler, the same advice. Up! The world (perhaps you now look upon it with pallid and disgusted eyes) is full of zest and beauty for you, if you approach it in the right spirit!"). "Manly Health and Training", The New York Atlas, September 12, 1858, 1. 14 A. L. Wagner, Adversaries of Dance, op. cit., p. 156-161. "The combination of overheated rooms and dancing to exhaustion, along with rich food consumed late at night and entry into chill air without proper clothes, regularly and inevitably, in the minds of many dance opponents, produced disease and death. Franklin Wilson coined the epitaph to this line of reasoning: 'The dance has proved to them (ladies) the herald of death, and the ball-room the gate of the grave'" (Ibid, p. 161). 15 (while people from the working class were thought to remain vigorous and healthy because they were physically active)-which, as Killinsgworth argues, most certainly influenced Whitman's positive outlook on physically active bodies. [START_REF] Killingsworth | Whitman's Poetry of the Body[END_REF] For Killingsworth, "Whitman's physicality implies a new understanding of beauty based on the health and vigor of the body." 17 And what he calls Whitman's "physical morality" would later in the century be echoed by the principles of François Delsarte, which were to become increasingly popular in the second half of the nineteenth century in America. 18 For Delsarte and his disciples, the body doesn't lie, and each emotion is echoed and expressed through the body-which should never be repressed by moral or bourgeois conventions and should be free to unleash its expressive power. [START_REF] Steele Mackaye Gave | 1870s popularizing Delsarte's theories among the American public[END_REF] A similar class bias is perceptible when Whitman mentions foreign dances: in these poems, the waltz or the polka that were all the rage in genteel ballrooms in the second half of the nineteenth century, are replaced with less formal, free and energetic foreign dances, like the Irish jig or Spanish dances. [START_REF]section 3, LG, p. 341) and to "the Spanish dance with castanets in the chestnut shade[END_REF] When Whitman does evoke the waltz, as in section 4 of "Proud Music of the Storm," (LG, p. 342-343), it is part of a cosmopolitan whirl of dances and music, where opera is heard along dance music and "the martial clang of cymbals," and where all religious dances are represented, from "dervishes," "the rapt religious dances of the Persians and Arabs," "modern Greeks dancing," to "wild old Corybantian" dances. The exultation of the body in these religious dances carries erotic undertones, which are emphasized by the cyclical, revolving rhythm of the section: the anaphoras "I see," "I hear" set the lines in a circular movement which always comes back to its beginning, thus breathing in considerable dynamism into the section as well as placing all these dances on the same level, with no hierarchy whatsoever. The section creates an image of a cosmopolitan whirl of dancing bodies, all united in this universal dance: the worshippers pray with their bodies, and through the music accompanying these dances; they offer their body in worship, an image that culminates in that of the last dancers evoked in the section, the Indian "bayaderes." The reference to the Indian "bayaderes" leaves no doubt as to the extent of Whitman's dance culture: appearing steadily in European ballets from the seventeenth century onwards, the character of the "bayadere" had become one of the great romantic heroines on the European stages, along with the characters of the Sylph and Giselle. [START_REF]One of the first apparitions of this character dates back to Lully and Beauchamps[END_REF] Extracts from the ballet Le Dieu et la Bayadère, choreographed by Filippo Taglioni, were danced in America from the 1840s, in the wake of the American tour of the famous ballerina Fanny Elssler. The character of the bayadere is inspired by the Indian devadāsī ("servants of the gods," in Sanskrit) who were both courtesans, dancers and sacred servants of the temple. These characters, who arrive last in the enumeration of all the sacred dancers in the poem, crystallize this combination of sacredness and eroticism, the climactic exultation of the body in dance.

These sacred dances can also be read in the light of Whitman's primitivism, a desire to revert to a form of immediate, true expression, to connect to the truth of human nature and to the truth of the body. 22 This is especially true when Whitman evokes Native American dance or movement. The "Red Squaw" who appears fleetingly in "The Sleepers", for example, is primarily evoked through her physical appearance, with particular emphasis placed on her physical fitness: "Her step was free and elastic, and her voice sounded exquisitely as she spoke. / My mother look'd in delight and amazement at the stranger, / She look'd at the freshness of her tall-borne face and full and pliant limbs" (LG, p. 361). In "Our Old Feuillage", Native American dance appears as part of the great American pageant, "the acts, scenes, ways, persons, attitudes of these States," -Whitman's sweeping presentation of America and its inhabitants as a huge life-size drama -in the form of "the drama of the scalp-dance enacted with painted faces and guttural exclamations" (LG, p. 94-95). The reference to "drama," and the pageant-like structure of the section that features Native American dance, recall the structure of nineteenth-century theatrical performances on American stages. These were organized in a succession of scenes where different forms of entertainment (dance, musical interludes, plays, …) were offered to the spectators in the space of one evening. 23 In that sense, Native American dance is transformed into a symbol of primitive Americanness, of a naturally embodied Americanness, moving freely and expressing itself physically-the starting point of the great American pageant. 24 Dance in Leaves of Grass thus provides significant insight into Whitman's representation of contemporaneous issues of class and national identity. It also plays an important part in his representation of gender and eroticism. In that perspective, physical interactions in "The Sleepers" can be read as a dance in which the lyric self emerges as protean as ever, containing the proverbial Whitmanian multitudes.

II. Lyric choreography in "The Sleepers"

In Le Corps et sa danse, Daniel Sibony proposes that the ways the body carries itself, interacts with others and with the world around it, can all be translated in terms of dance: le monde est de l'énergie qui danse : qui cherche au hasard des passages marquants, des mutations fécondes. [ 23 In The Portable Theater, American Literature and the Nineteenth-Century Stage (Baltimore and London, The Johns Hopkins University Press, 1999), Alan Ackerman delineates how performances on nineteenth-century American stages featured settings that were typically American, and, quoting Mary Ryan, explains that "between 1825 and 1880, [Americans] invented the modern parade." He further explains that "between these years the parade in America signified much about American democracy, documenting 'the development of such concepts as class, ethinicity and gender'" (p. 28). For further discussion of the relations between the stage and American culture in Whitman's time, see p. 1-41. 24 A few decades later, the pioneers of American dance would also turn to Native American traditions and dances in the search of a primitive, true American movement, and would incorporate Native American dances into their own projects of choreographing America, their own mise en scène of the great American drama. One of the best examples is Martha Graham's Primitive Mysteries (1931). Graham also wanted to include an "Indian Girl" in Appalachian Spring (1944). 25 Daniel Sibony, Le Corps et sa danse, Paris, Seuil, 1995, p. 94-95.

The enumeration of dances in the 4 th section of "Proud Music of the Storm" is pure "énergie dansante," to use Sibony's terms, energy in motion: the circular rhythm takes the addressee in this dance around the globe with the speaker. A similar use of a circular rhythm, triggering a circular movement which draws the addressee in a whirl of moving, dancing bodies, can be found in "The Sleepers," (LG, p. 356-364) in which Whitman develops a particular form of lyric choreography. In this poem, the speaker conjures visions of bodies in different states, sleeping bodies, dead bodies, bodies engaging in physical activities, and celebrates the beauty and strength of the human body-from the vigor of the "beautiful gigantic swimmer" and his white muscular body battered by the waves in section 3, to the "pliant limbs" of the "red squaw." The universal beauty of the human form, its health, are once again celebrated: "Perfect and clean the genitals previously jetting, and perfect and clean the womb cohering, / The head well-grown proportion'd and plumb, and the bowels and joints proportion'd and plumb" (LG, p. 362). The poem reads like an endless whirl of bodies, conjured up and materialized through the lyric voice, in the circular movement and rhythm of an informal waltz, in which men and women waltz in and out, and the speaker-whose body's contours are frequently blurry, as his body merges with others'-appears in the middle of this whirl.

The body in "The Sleepers" is essentially protean and dynamic, capable of endless shifts in shape, appearance, and gender ("I am the actor, the actress, the voter, the politician" LG, p. 358). This protean quality intimately connects the lyric and dance. As Daniel Sibony argues, dance is an art form questioning the body's limits, an art form that, like the lyric, is essentially destined to an addressee-visible or invisible-and questions the origins and limits of selfhood: "la danse questionne toutes les façons qu'a le corps de se porter vers les frontières, des limites d'être, des mouvements originaux -qui mettent en jeu l'origine."26 "The Sleepers" equally questions the limits of selfhood, since the protean speaker has the capacity not only to touch, but also to inhabit other bodies, to experience touch, physical contact from their own bodies. Not unlike dance which, according to Sibony, perpetually reverts to the origins of the body, by constantly reenacting the birth and rebirth of the self to the world, Whitman's poem dramatizes the origins of the body, with the image of the "newborn emerging from gates, and the dying emerging from gates" in the first section. The poem repeatedly crosses thresholds of selfhood and physical experience (being born, dying, sleeping and awakening, motion and stillness, the fusion of two bodies in sexual intercourse) and transcends the limits of the body to celebrate the universality of being, the vitality and energy of the human body and of the lyric. The lyric act of speaking to a faceless addressee becomes a dance to a faceless, universal public, in which the lyric self, at the center of this whirl of bodies that he bodies forth through language, becomes pure lyric energy-he becomes the dance: I go from bedside to bedside, I sleep close with the other sleepers each in turn, I dream in my dream all the dreams of the other dreamers, And I become the other dreamers. I am a dance-play up there! The fit is whirling me fast! (LG, p. 357)

In "Hair, Feet, Body and Connectedness in 'Song of Myself'," Taylor Hagood argues that one way of "understanding Whitman's complicated presentation of 'individuality vs democracy' lies in his depiction and positioning of body in 'Song of Myself.'"27 He further contends that "head and feet" represent points of contact thanks to which the body can appear as a solidified entity or, on the contrary, as points between which the body dissolves:

These points (head and feet) may be the boundaries of a fluid and consolidated entity, or they may represent extremes that serve not to solidify but rather to fragment and disconnect the body and the self. Connectedness and consolidation embody Whitman's democratic ideal and are attainable only when the head and feet successfully make contact with each other and with the earth. 28 In "The Sleepers," haptic connectedness is established in a similar way, through the hands of the speaker touching other selves, and his feet, evoked in the opening lines of the poem:

Stepping with light feet, swiftly and noiselessly stepping and stopping, Bending with open eyes over the shut eyes of sleepers, Wandering and confused, lost to myself, ill-assorted, contradictory, Pausing, gazing, bending, and stopping. The speaker's all-encompassing vision allows him to see every sleeper, to move from bed to bed, but vision is soon replaced by touch, from the soothing touch that helps the sleepers go back to sleep ("I pass my hands soothingly to and fro a few inches from them, / The restless sink in their beds, they fitfully sleep" [LG, p. 357]) to erotic touch, when the speaker becomes a woman awaiting her lover and finally reunited with him in physical love (LG, p. 358): I am she who adorn'd herself and folded her hair expectantly, My truant lover has come, and it is dark. Double yourself and receive me darkness, Receive me and my lover too, he will not let me go without him. I roll myself upon you as upon a bed, I resign myself to the dusk. He whom I call answers me and takes the place of my lover, He rises with me silently from the bed. Darkness, you are gentler than my lover, his flesh was sweaty and panting, I feel the hot moisture yet that he left me.

A gender shift is performed in this passage, in which bodies merge, undercover of the engulfing darkness: the bodies of the lovers dissolve into one another, into darkness, and the only haptic memory left is the residual sweat of the male lover on the speaker's skin. 29 Physicality is spectral, materialized in the remnants of a touch, these traces of "moisture," the haptic memory of flesh upon flesh. The contours of the speaker's body are fundamentally labile, moving from male to female, merging with the lover's body in the sexual experience, carrying the ghosts of previous incarnations within itself. 30 In that sense the speaker (and speaker's body) of this poem is a perfect illustration of the lyric condition that can only be embodied through effects of voice. Between the two points of contact of the head and feet, the lyric self dissolves; he is not a dancer, but the dance itself as proclaimed at the end of section 1, dancing energy in perpetual motion. This mode of haptic connectedness and this representation of the self as a blurry center of energy in between the two connecting points is also what connects dance and the lyric. This dichotomy between an elevation, a projection upward and outward and a firm grounding of the feet on the supporting ground is the essential animating principle of dance-whether one refers to the classical dance treaties by Noverre or Blasis [START_REF] Noverre | Lettres sur la danse[END_REF] , or the more contemporary conceptions of dance. It is also what defines the lyric; according to Jean-Michel Maulpoix, the lyric deploys itself between its starting point, the origin of speech, and this ideal of elevation. For Maulpoix, "il n'est rien de plus essentiel à la définition du lyrisme que l'idée d'élévation. Elle en est le principe actif." 32 He adds: "le lyrisme regarde du côté de la source du poème (autant que son embouchure, ou de sa mise en voix), c'est-à-dire là où les énergies affluent et aspirent à se dépenser." 33 Dance, like the lyric, can only thrive in this perpetual in-betweenness, between the anchoring ground and a higher point, in this spatial tension between two opposite points, and between two bodies, the lyric self's and the addressee's. If, following Maulpoix still, the lyric thrives in liminal spaces it is also the case for dance. 34 For Daniel Sibony, "la danse déploie dans son espace la question de l'entre-deux-corps, le jeu des places et de leurs genèses, de leurs engendrements." [START_REF] Sibony | [END_REF] In "The Sleepers", the lyric self is perpetually "entre deux corps," moving between two bodies, two incarnations-ever liminal.

A few decades after the composition of "The Sleepers," French poet Stéphane Mallarmé would write about American dancer Loïe Fuller's performance that dance, like lyric poetry, is fleeting, mutable and evasive, evoking the "vibrant suspension" (suspens vibratoire) of a dance when the dancer is no longer a woman but a "sign," detached from the constraints of personality and individuality and free to be as protean as the lyric self. [START_REF] Mallarmé | Elle te livre à travers le voile dernier qui toujours reste, la nudité de tes concepts et silencieusement écrira ta vision à la façon d'un Signe, qu'elle est[END_REF] Commenting on Mallarmé's representation of the dance of Loïe Fuller, Mireille Ruppli and Sylvie Thorel-Cailleteau write:

Marquée par "l'inindividuel" ou "l'impersonnalité," elle (la danseuse) est désignée comme un "être dansant," "emblème, point quelqu'un" ; par là s'indique comment, dans un même mouvement, le mot perd sa dimension numéraire et l'auteur sa présence élocutoire : le sol où s'appuie l'envol de la danseuse est aussi miroir, par lequel s'opère la transmutation d'un être en signe, de la même façon que l'installation des mots sur la page réalise l'abstraction de la personne du poète. [START_REF] Ruppli | Mallarmé, la grammaire et le grimoire[END_REF] The dance in "The Sleepers" articulates this Mallarmean "Entpersönnlichung" of the lyric self, which becomes impersonal, evading the limits of selfhood, and becoming open to all forms of mutations, shape-shifting and gender-shifting. [START_REF]Aimé Césaire et la 'quête dramatique de l'identité': le Cahier d'un retour au pays natal[END_REF] This escape from personality may be one of the most modern aspects of Whitman's lyric.

Whitman's lyric self is therefore complex, at once embodied and physical and always attempting to merge into a greater, universal entity-personal and impersonal. The erotic experience is maybe one of the most telling examples of the lyric self's transgression of gender and bodily limits. Many critics have remarked that when homosexual sexual acts are evoked in Leaves of Grass, a gender shift is performed. [START_REF] Killingsworth | Whitman's Poetry of the Body[END_REF] In section 11 of "Song of Myself," the speaker operates a similar gender shift to the one performed in "The Sleepers." In what Michael Moon has called a "grammatically-transvestite moment," dance is associated to this liberation from the limits of gender and selfhood, as the self "passes through feminine identity," and enters the scene dancing. [START_REF] Moon | Disseminating Whitman[END_REF] "Dancing and laughing along the beach came the twenty-ninth bather, / The rest did not see her, but she saw them and loved them" (LG, p. 34).

Examining dance in all its forms-social dance, but also ballet-can shed new light on the representation of lyric identity as fundamentally labile and on the performance of gender in Leaves of Grass. When considering the occurrences of dance cited above, it becomes apparent that Whitman frequently refers to men dancing, or uses gender-neutral terms such as "partners" or "dancers," which tones down the female presence in the dancing group. The "midnight orgies of young men" and the dance in "Native Moments," like the dance between the two men in "We two Boys Together Clinging" are unusual in the context of the nineteenth century, since dance writings (dance treaties, anti-dance treaties, dance manuals, press articles about dance…) overwhelmingly focused on female dancers. When male dancers were mentioned, the focus was mainly on their manners (or lack thereof), not on their dancing-whereas the elegance of women's dress, carriage and dance was the main focus. Men dancing-especially amongst themselves, such as the young men in "Our Old Feuillage"-were seldom represented. It is even more true when we consider ballet: in what was called "the age of the ballerina," the male dancer was almost invisible, self-effacing so as to leave the centre of the stage to the female dancer. What is striking, therefore, in the lyric choreography that is performed in "The Sleepers," is that it is modeled not on the ballerina's part-as it would be expected in the nineteenth-century context-but on the male dancers' parts in Romantic ballet.

Critics like Robert Faner or Carmen Skaggs have highlighted the importance of the stage and the performing arts in the formative years of the poet, who attended the theatre regularly. [START_REF] Faner | Overtones of Opera in American Literature from Whitman to Wharton[END_REF] In fact, Whitman was a fixture at the New York theatres and opera houses. The great ballerinas of the 1840s Fanny Elssler and Augusta Maywood are mentioned in a letter sent to Whitman by Edward Mawson in 1885, and in his Reminiscence of the Bowery Theatre, Whitman recalls having seen Madame Céleste, one of the first famous French dancers to tour the country. 42 In an October 1862 article published in the Brooklyn Standard, Whitman also refers to social dancing (which he actually engaged in) and a dance inspired by La Bayadere-thus confirming the fact that he did know the ballet. [START_REF]Brooklyniana[END_REF] If we consider the character of the Spanish dancer in Whitman's recently (re)discovered Life and Adventures of Jack Engle, the influence of the ballet on the poem becomes undeniable. Fanny Elssler, the Austrian ballerina who toured the country in the early 1840s, specialized in Spanish dancesla cachucha being her specialty. Elssler had a huge impact on American audiences, especially on men: Emerson mentions her in his journals, while Longfellow discusses her in his correspondence, even basing the character of Preciosa in The Spanish Student on Elssler. Given his admiration for strong, healthy, active bodies, Whitman would definitely have admired the graceful and vigorous dancing of ballerinas like Elssler, but it is also very likely that he should have been inspired by the male dancers who partnered Elssler or Maywood on stage.

In "The Sleepers" the central position of the self, coupled with an effacement of his body to the benefit of other bodies, and the circulation of his gaze echo the situation of the male dancer on stage. The representation of the speaker at the center of the poem, watching the "nimble ghosts," the whirl of bodies around him as he steps lightly across space, is common to many romantic ballets, where the lead male dancer often finds himself surrounded by whirling, dancing bodies. To wit the many scenes by the lakeside in Swan Lake where Siegfried watches the swans dance before engaging in a pas de deux with Odette, or the Shadows ("Le royaume des ombres") in the last act of La Bayadère, when Solor is deep in opium-induced sleep and the bayadères dance around him, or in the two ballets that were popular during Whitman's formative years, La Sylphide and Giselle. In the second act of Giselle, Hilarion and Albrecht also find themselves at the center of the whirl, when the wilis dance around them. In the second act of La Sylphide, James also adopts the same posture, when he is in the forest and all the sylphs dance around him. Those two scenes where the male dancer is surrounded by ethereal female dancers, the ghosts of abandoned maidens, may have provided inspiration for the speaker's vision of the "nimble ghosts" dancing around him in section 1.

The speaker, it should be noted, sees everyone, but is not looked at, except, one may surmise, in the sexual act depicted at the end of the first section, when the speaker becomes the woman waiting for her lover. This circulation of the gaze is typical of the situation of the male dancer on stage: his role is to partner the ballerina, to show her off, and his presence is deliberately upstaged by his female partners (this convention in classical ballet only changed in the late nineteenth century with Petipa, and even more in the following century, with Nijinsky and Nureev, who completely modified the role and our vision of male dancers). In the nineteenth century, the male dancer looked at his partners, but he was not meant to be looked at. The gaze was always directed to the female dancer. Dance critics have repeatedly shown how Romantic ballets were choreographed and staged for a principally male heterosexual audience, with a male heterosexual gaze in mind. The fact that Whitman models his lyric choreography on the male dancer's part-being in a central position, becoming the recipient of gaze and touch only when the self becomes feminized-is extremely significant and transgressive. First it attests to Whitman's anti-bourgeois perspective on ballet. Indeed, ballet critics of the nineteenth century often depicted male dancers negatively: they loved to see the female form displayed on stage, but shrunk from the spectacle of the male body. In 1838, Théophile Gautier wrote: "rien n'est plus abominable qu'un homme qui montre son cou rouge, ses grands bras musculeux, ses jambes aux mollets de suisse de paroisse, et toute sa lourde charpente virile ébranlée par les sauts et les pirouettes." 44 According to Ramsay Burt, "For the mid-nineteenth-century bourgeois ballet critic, vigorous and manly displays of Theatre, the Bowery Theatre and the National Theatre. American ballerina Augusta Maywood made her debut as "La Bayadère" on the Philadelphia and New York stages before leaving America for Europe. 44 Théophile Gautier, OEuvres Complètes, Critique théâtrale, T1, éd. P. Berthier et F. Brunet, Paris, Honoré Champion, 2007, p. 487. dancing might sometimes have carried negative connotations of working-class entertainment." 45 That Whitman does not shy away from representing the lyric self as a male dancer can partly be explained by his affinity with the working class and his desire to represent the body as graceful because it is strong and vigorous, not according to traditionally bourgeois representations of the body. This construction of the lyric self as a male dancer is typical of Whitman's working-class erotics, since the strong, physically active body, is desirable 46 , but also because of the fluidity in gender identification in the circulation of the erotic gaze: the passage through female identity in "The Sleepers" or the subversion of the male heterosexual gaze in Whitman's lyric choreography intimately associates Whitman's working-class erotics with the notion of mobility. As Jimmie Killingsworth notes in his study of "The Sleepers," sexual initiation is associated to the working-class. I would add that it is the working-class's inherent mobility (present in the name of the initiators, "the journeymen" 47 ) which allows for the homosexual experience to take place-just as gender fluidity allows for the sexual encounter with a male partner to take place in the same poem, or in the episode of the "twenty-ninth bather" in "Song of Myself." This association of the working-class with some form of feminization and mobility is significant in its nineteenthcentury context: Michel Foucault, Elaine Showalter or Janet Beizer for example have shown that the working-class was equated to instability and mobility in the nineteenth century (whereas the bourgeois class was associated with stability), and thus to a form of social and sexual deviance and deviation. 48 The affinity with working-class bodies and the deviation of the erotic gaze in Whitman's lyric choreography is a way to encode the homosexual experience, since for Ramsay Burt, looking at the male dancer in the context of nineteenthcentury ballet is necessarily a homoerotic act:

To enjoy the spectacle of men dancing is to be interested in men. Because there was no acknowledged distinction between ballet as aesthetic experience and ballet as erotic spectacle, … the pleasures of watching men dancing became, in the mid-nineteenth-century, marred by anxieties about masculine identity. 49 45 Ramsay Burt, The Male Dancer, Bodies, Spectacle, Sexualities, NY & London, Routledge, 1995, p. 27. 46 In the previously quoted "Manly Health and Training" series, Whitman defines the "perfect body" has a body with "perfect blood-no morbid humors, no weakness, no impotency or deficiency or bad stuff in him; but all running over with animation and ardor, all marked by herculean strength, suppleness, a clear complexion, and the rich results (which follow such causes) of a laughing voice, a merry song morn and night, a sparkling eye, and an ever-happy soul!", The New York Atlas, September 12, 1858, 1. In other articles from this series, Whitman continues to emphasize the necessity for physical exercise outdoors and the beauty of the vigorously active man's body, while the sedentary middle-class or upper-class body is perceived as weaker and more prone to the above quoted "morbid humors". 47 According to Killingworth, "'Journeymen' signifies members of the working class, but a special breedskilled, accomplished workers, special companions for the sexual neophyte, from whom 'they hide nothing'" (Whitman's Poetry of the Body, p. 17-18). 48 See Michel Foucault, Histoire de la folie à l'âge classique, Paris, Gallimard, 1976; Elaine Showalter, The Female Malady: Women , Madness and English Culture, 1830-1980, New York, Pantheon Books, 1985; Janet Beizer, Ventriloquized Bodies, Narratives of Hysteria in Ninetenth-century France, Ithaca, Cornell University Press, 1994. Commenting on an excerpt from Michèle Ouerd's Leçons sur l'hystérie virile ("la classe ouvrière est imaginée dans le grand corps social de la République fin de siècle comme l'utérus migrateur de l'hystérique traditionnelle" [Paris, Le Sycomore, 1984, p. 27]), Beizer writes: "By the latter part of the century the hysteria label was an attempt to pin down or arrest the upwardly mobile desires of various social outcasts. These marginalized elements come to include not only those perceived as inferior by sex or by class, but by race as well. (…) Typed as unsettled, undisciplined, and unstable, the unempowered classes and races are effeminized, and hystericized by analogy to woman" (Beizer, op. cit., p. 50, ellipsis mine). 49 J. Burt, The Male Dancer, Bodies, Spectacle, Sexualities, op. cit., p. 28.

For Whitman,"being a dance" ("I am a dance" [LG, p. 357]), I would eventually argue, equals being a protean lyric self, performing endless shifts in shape, appearance, gender and social class, a self, connected through the feet and the head to the two cardinal points of the earth and the sky, the beginning and the end of the poem, but ever-mutable in-betweenembodied, personal and impersonal at the same time. The lyric is the expression of a desire to go beyond the borders of selfhood, to venture forth in the world through language, a desire which also applies to dance. Just as nothing "happens" in a poem but the birth of the lyric self to the world, the projection of the lyric utterance, nothing happens in a dance but the body's awakening and its expression, the projection outward of the body's energies. Languagephysical or verbal-is energy for Whitman, and always celebrates, performs, the vitality and glory of the human body and human experience. If Whitman "make(s) words dance," he also creates this unique lyric choreography for a self whose physical contours appear very strongly, and then dissipate. 50 In his poems, dance appears as a liberating physical experience, allowing the self to escape personality and simultaneously, if paradoxically, be extremely physical and embodied. Whitman's poetry, through and as dance, thus challenges nineteenth-century bourgeois expectations regarding class and gender and creates a dynamics of mutation and movement which allows for the homoerotic experience to take place surreptitiously, among the many shifts in bodies, gender and perspectives. 50 In An American Primer, Whitman writes: "A perfect writer would make words sing, dance, kiss, do the male an female act, bear children, weep, bleed rage, stab, steal, fire cannon, steer ships, sack cities, charge with cavalry or infantry, or, do anything that man or woman or the natural powers can do" (p. 16).

  See H. Aspiz, Whitman and the Body Beautiful, T. Nathanson, Whitman's Presence: Body, Voice and Writing in Leaves of Grass, M. Moon, Disseminating Whitman: Revision and Corporeality in Leaves of Grass, J. Killingsworth, Whitman's Poetry of the Body; Sexuality, Politics, and the Text.

  …] [L'énergie] dansante fait danser l'entre-deux, les formes passagères et transitoires ; la danse traduit la masse corporelle en énergie, et l'énergie en masse. Le corps est une réserve d'énergie dansante, toute impliquée dans la naissance au « monde », la naissance du monde, dont le danseur fait partie comme il fait partie de sa danse. 25 22 See for example Helen Carr, Inventing the American Primitive: Politics, Gender and the Representation of Native American Literary Traditions, 1789-1936, New York, New York University Press, 1996, p. 220-222; Yvor Winters, Primitivism and Decadence: A Study of American Experimental Poetry, New York, Haskell House, 1969, or W. Pannapacker, "'The Bricklayer shall lay me': Edward Carpenter, Walt Whitman, and Working-Class 'Comradeship'."

D. Sibony, Le Corps et sa danse, op. cit., p. 10.

T. Hagood, "Hair, Feet, Body and Connectedness in 'Song of Myself'", p. 25.

Ibid., p. 25.

J. Killingsworth notes that darkness in this passage is "permissive," allowing the homosexual initiation to take place, hidden by darkness and the gender shift (Whitman's Poetry of the Body, p.[START_REF] Steele Mackaye Gave | 1870s popularizing Delsarte's theories among the American public[END_REF][START_REF]section 3, LG, p. 341) and to "the Spanish dance with castanets in the chestnut shade[END_REF].

For Jean-Michel Maulpoix, "le sujet lyrique est un sujet plein de voix tues qui sont comme les dépouilles de ses chimères et de ses potentialités" ("La quatrième personne du singulier, esquisse de portrait du sujet lyrique moderne," in Figures du sujet Lyrique, éd. Dominique Rabaté, Paris, Presses Universitaires de France, 1996, p. 159).


Hervé Théry
email: hthery@aol.com

François-Michel Le

Tourneau Credal

Population and environment in Brazil

Keywords: Brazil, Population, environment, sustainable development, Amazonia

Brazil is a continent-sized county, with a great diversity of ecosystems, unequally transformed by the processes of occupation and territorial formation. In the southern half of the country, the natural environment has been replaced by a humanized natural environment due to the pushing back of settlement frontiers, which, for five centuries, has transformed Brazil's land use. In recent decades the rhythm of change has been faster: on one hand, the population has increased, and migrated to the cities; on the other hand, the cultivation of major crops (coffee, sugar, soybean, cotton, etc.) has moved northwards and westwards, reaching northern regions such as Mato Grosso and Pará, threatening the rainforest in the Amazon Basin.

Brazil, a continent

Brazil is a continent-size country; it extends from north of the Equator to south of the Tropic of Capricorn and from the 36 th meridian to the 72nd west of Greenwich.

). A car driver, or more often a truck-driver, who travels to the city posted on the sign knows that one can expect to spend hours, even days, on the road to reach it.

It was by no means always obvious that Brazil was to become the giant it is today: the slice of South America allocated to Portugal by the Treaty of Tordesillas (which

delimited Spanish and Portuguese possessions in 1494) was marked by a meridian passing by the mouths of the Amazon. Two and half centuries later, the current boundaries, are nearly everywhere three thousand kilometers further west. Strong nationalist sentiments forged during this conquest made it possible to overcome social and regional cleavages and create the huge country (Figure 2).

Paradoxically, Brazil mainly owes this immense territorial extension to the pressure of foreign competitors. These forced Portugal to take seriously an unwillingly engaged conquest, then to complete and consolidate it. Without the competitors, the country may not have achieved the same destiny, but this competitive pressure does not explain everything. The leading factor in the extension and unification of the Brazilian territory is, rather, the coincidence of deliberate political action by the Portuguese Crown (and later by the Brazilian governments) and of the impressive pioneering spirit of Brazil's inhabitants. This continental space was indeed conquered, built, and consolidated step by step (Théry and de Mello 2003).

Figure 1. Road distances

The largest photo was taken in Santarém (Pará). Secondary photos were taken on the road from Brasília towards the north (top left), at the Brazil-Venezuela border (top right), in Acre State (middle left and between Brasília and Goiânia (bottom left).

(Photographs by Herve Théry)

The territory we see today bears the mark of this history; it has been able, with a few adjustments, to accommodate the evolution of a country whose population has increased tenfold in the last century (from 17 million inhabitants in 1900 to nearly 170 in 2000).

New muncipalities (municípios) were constantly created in the newly conquered regions (and some are still being created), new states were founded (and some Brazilians would create yet more) but overall the structure built in the first centuries of Brazil's life still holds.

Figure 2. Brazil's size compared to Europe

Population densities and their evolution

The Brazilian population is unevenly distributed on the territory; there is still a sharp contrast between coastal and inland regions, which still reflects the effects of the colonization process and settlement of the territory. The former are highly populated, while the latter are only marginally occupied (Figure 3).

Even areas of population concentration are irregular: in populous states like Rio Grande do Sul, Santa Catarina, Minas Gerais, Bahia, Pernambuco, Paraíba and Rio Grande do Norte, there remain many interspersed, thinly populated areas. Only Sao Paulo, Paraná, Rio de Janeiro, Sergipe, Alagoas, Paraíba and Ceará have a continuously occupied territory. In the rest of the country, the population distribution is closely correlated with the transport networks ─ waterways, railroads (built to export the products of the inland regions) and now regional feeder roads The distribution of the population shows therefore a clear east-west opposition, the result of the process of occupation and colonization starting from the coast (Figure 3).

So in the parts closest to the sea, Nordeste, Sudeste and South, the densities exceed 50 inhabitants per square kilometer, and can rise to over 10,000 in the state capitals. In contrast, in most of the Amazon and in large areas of the Midwest, densities are very low, between 0.13 and 10 inhabitants per square kilometer. They are only higher in the state capitals and some municípios, where they are between 11 and 24 inhabitants per square kilometer. In fact, Amazon densities, calculated for the whole municipality, have little meaning, as a great part of them is composed of forests, but the high densities of the coast and the contrast with the interior regions remains one of the strong features of the country.

The coastal area is not homogeneously occupied: nearly empty north of the Amazon, there is a clear contrast between two highly populated regions, on both sides of a sparsely occupied center (southern Bahia State and Espírito Santo State). In the Nordeste, the contrast between the coastal zone and the interior is repeated. In the Sudeste and South, however, the density remains high in many regions close to the western borders of the country. It is the only place where Brazil has some "depth", as shown on the density map between the latitudes of Vitoria and São Paulo, and between the sea and the edge of Mato Grosso, Brazil's agricultural, industrial and urban heartland.

Figure 4. Population densities by countries and Brazilian States

The population densities contrast so much inside Brazil that its states are, from this point of view, similar to countries with extremely different situations (Figure 4). The Federal District of Brasília (421 inhabitants per km 2 ) is similar to the Netherlands (395) or to Israel (368). São Paulo ( 160) is a little more densely populated than Poland (123), Nigeria (139) and China (137). Pernambuco (85) is at the same level as Spain (80) or Turkey (89), and Minas Gerais (32) as United States (31) or South Africa (36). Among the least densely populated states, Goias ( 16) is similar to Argentina (14) and Algeria (13), Pará (5) to Kazakhstan (5.5), and Amazonas (2) to Canada (3.2) and Libya (3.3).

How does this distribution of the population change, and how do these changes affect the environment? The IBGE (Instituto Brasileiro de Geografia e Estatística) has recently published the results of the Population Count of 2007 (Diário Oficial da União of October 5, 2007). According to these data, Brazil currently has 183.9 million inhabitants. In the seven years after the 2000 census, Brazil's population grew by more than 14 million inhabitants, the equivalent of Bahia State, Brazil's fourth most populated state. All major regions grew, but without changes in the hierarchy: the Sudeste is still the most populous area, with 77.8 million, followed by the Nordeste (51.5), South (26.7), North (14.5) and Midwest (13.2). Among the states, Sao Paulo leads with 39.8 million inhabitants, followed by Minas Gerais (19.2), Rio de Janeiro (15.4), Bahia (14 ) and Rio Grande do Sul (10.5). The least populated is Roraima (395.7 thousand inhabitants), followed by Amapá (587.3 thousand) and Acre (655.3 thousand).

Figure 5 shows the regions that gained (blue circles) or lost population (red circles).

The former are obviously much more numerous, owing to the country's population increases, which also increased the dominance of the coastal regions and major cities.

Loss of population affected some regions, such as Rio Grande do Sul, southern Bahia, north of Paraná and ─ most surprisingly ─ the center of Rondônia and Amazonas, till recently pioneer regions which attracted migrants from other areas. The creation of new municipalities in 2001 may explain some cases, especially in Rio Grande do Sul, but in the other cases there was a real decline in population, a novelty in the demographic history of the country. How can we explain those contrasted movements? One of the main explanations is to be found in the shifting distribution of agricultural regions. [START_REF] Grégoire | L'ogre et le petit Poucet (The ogre and Tom Thumb)[END_REF]. The best example is without contest the case of soybeans. Brazil did not produce soybeans until the 1970s, and it is now the world's second soybean seed producer, the largest exporter of soybean meal, and one of the leaders for the oil. This increase was achieved through the cultivation of the cerrado savannas in Mato Grosso, Goiás and west of Bahia, while in the "old" regions (those with production in the 1970s) soybean was displaced by other products (Figure 6) (Théry 2004). where it is produced for commercial purposes), which is associated with the pioneer frontier, has followed the same pattern as soybean production. Rice production has therefore being increasing along the arc of deforestation and declining behind it, where it has been replaced by other crops, or more frequently by grass for cattle. Cotton has followed roughly the same pattern as soybeans, and in this case Mato Grosso has also become the largest domestic producer [START_REF] Bertrand | Le marché mondial et l'expansion du complexe soja dans les cerrados du Mato Grosso (Worldwide market and the expansion of the soya complex in the cerrados of Mato Grosso)[END_REF]. Northward expansion of cattle raising made it possible to constitute immense herds in the new areas, and to nourish those in the older areas. As a result, there are today more cattle than human beings in the country, although the Brazilians also annually consume the equivalent nearly 30 million animals (one per annum for each six people). In addition, benefitting in particular from the European difficulties related to the "mad cow" crisis, Brazil became in 2003 the world's leading exporter of beef.

In the ten years between the 2006 census from and the precedent, the growth of the livestock (Figure 7) occurred in the whole of the North (except for the island of Marajó) whereas the few declining areas are concentrated mainly in the Southeast (particularly the state of São Paulo). The dark green areas on the map, indicating the strongest positive variations, clearly underline the arc of deforestation which runs from Maranhão to Acre and even in areas located (more worrying still) in front of that arc, including the state of Amazonas, which has not yet been affected by deforestation.

Figure 8. The transformation of Brazil's ecosystems

The impact on the environment The expansion of cattle breeding is one of the main causes of the fast deforestation impacting the country. Figure 8 shows the extent of the transformations that have occurred during the forty last years. The "anthropized zones" ( in their vast majority, cleared) between 1960 and 1997 show the continuity, but also the acceleration of the processes of occupation that started with the arrival of the Portuguese on the littoral, five centuries ago [START_REF] Dean | With Broadax and Firebrand: The Destruction of the Brazilian Forest[END_REF]. The deforestations of the current era affect primarily the cerrados savannas and Amazonia, incorporated by the pioneers fronts into the agropastoral complex. This conquest of the frontier has been encouraged, or even caused, by instituting public policies (credit, infrastructure, etc.) which started the surge of the migrants and transformation of the landscapes.

The most heavily impacted zone forms an immense triangle, whose base is located in the west of Paraná and extends to the Northwest to Rondônia and the Northeast to Maranhão and southern Pará, leaving aside (for the moment?) the Pantanal swamps.

The conquest of northern Mato Grosso was incomplete at the end of the 20 th century but has appreciably progressed since then, and can be expected to continue in southwest Pará and southern Amazonas. The movement launched under the Presidency of Juscelino Kubsticheck, the promoter of "march towards the West", consumed a major ecosystem at each decade, the cerrados in the 1970s, the transition between cerrados and forest in the 1980s and in the 1990s it has clearly progressed into those. This process is followed by the Brazilian scientists of the INPE (National Institute of Space Studies), with more precision, in spite of the vastness of the territory concerned, than in many countries, where measurements of deforestation are made at 5 to 10 years intervals. In Amazonia it is measured annually, using 229 satellite images which cover the areas of primary forest, that is to say four of the five million square kilometers of Amazonia (Figure 9, Table 1). About 745,000 square kilometers (or 74,5 million ha, that is to say, 135% of the surface of France) have been cleared in Amazonia through 2009 according to the INPE, and with a very clear acceleration during recent decades. Furthermore, INPE cannot see, on the satellite images it analyzes, the selective cuts which result in a loss of the biodiversity and an aggravation of the fire hazard. One of the mains themes of these debates (which could in fact become a solution, at least a partial one) is the ecological and economic zoning of the area (zoneamento ecológico-econômico), which should be one of principal tools for town and country planning, with a clear definition of the zones to be preserved and those which can be exploited. But the divergences on methodology (which sometimes mask opposition to the very principle of zoning) slowed down or blocked in many cases agreement on land use planning mechanisms and their adoption and implementation. But in fact a kind of zoning has already been made, if one takes into account the zones already protected as parks, reserves and the indigenous lands.

On the whole, in most of Brazil, except in part of the Amazon forests, the natural environment has been replaced by a humanized natural environment. New transformations are to be expected, in northern Amazonia (still largely intact but increasingly integrated into the remainder of the country), as new roads (like the exit route towards the Caribbean, via Venezuela), and in the South, where new trade and transport activities related to the integration of Brazil in Mercosul upset established conditions. Obviously, debates on the use of the natural resources and on the condition of the environment in Brazil are not likely to die out in the near future.

Figure

  Figure 3. Population density

Figure 5 .

 5 Figure 5. Population gained and lost between 2000 and 2007

Figure 6 .

 6 Figure 6. Regional change in soybean production

  We are witnessing the rise of real pioneer settlement regions in Brazil, unparalleled in the contemporary world (except for certain parts of Indonesia or Malaysia), which recall those of the United States in the 19th century or, closer in time and space, those based on coffee in the states of São Paulo and Paraná in the 1930s and 1940s, masterfully analyzed by Pierre[START_REF] Monbeig | Pionniers et planteurs de São Paulo (Pioneers and planters of São Paulo)[END_REF].

Figure 7 .

 7 Figure 7. Spatial variation in the number of cattle

Figure 9 .

 9 Figure 9. Annual deforestation in Amazonia

Figure 10 ,

 10 Figure 10, Annual deforestation in the main Amazonian states

Table 1 . Total and annual clearing of forest in Amazonia

 1

	Year	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009*
	Clearing (km 2 )	18 226 18 165 21 523 25 396 27 772 19 014 14 196 11 633 12 911 7 008
	Change from prior year (%) 5,60	-0,33	18,49	17,99	9,36	-31,54 -25,34 -18,05 10,99	-45,72
	Total (millions ha)	587	605	627	652	680	699	713	725	738	745
	* Estimate. Source : INPE/Prodes,
	http://www.obt.inpe.br/prodes/prodes_1988_2008.htm

http://www.oeco.com.br/reportagens/37-reportagens/22886-o-mais-baixo-desmatamento-da-historia, consulted 14/2/2010

 Théry, H. 2004: La vague déferlante du soja brésilien.( The irresistible wave of Brazilian soya) M@ppemonde 74 (2). http://mappemonde mgm fr/num2/articles UPDATED NOTES 3/11/11:

1. The scale used for Fig. 7 is unclear, as is, therefore, some of the related text. How is the scale determined? Should any text be changed in the paper? Done in the text. The legend soul be changed to Variation in the number of cattle, from 30% to 89%, from 0% to 30%, from -20% to 0%, from -60% to 20%. Casey could you please do this change?

2. Shouldn't we add figure and table sources to the references list? If this is the case, I would need more information for some of the figures in order to inset references within the reference list. I don't think so, all of them already have references 3. Should references in non-English be translated and included in parentheses on the reference list? At least one other paper in the issue has done this. If so, can Herve (or someone) provide me or Casey a translation so that it can be put into the reference? 10: I believe that the scale on the left of the chart should read "ha/year".

Unfortunately not, it's km2/year 8. Note that I have renumbered several figures, and they differ a bit from Herve's last edition owing to an apparent numbering problem within it. Done.