Nicola Impollonia

Giuseppe Ricciardi

Fernando Saitta

VIBRATIONS OF INCLINED CABLES UNDER SKEW WIND

Keywords: Cable dynamics, Proper Orthogonal Decomposition, Wind forces

A non-linear finite element model of inclined cables, i.e. cables with non-leveled supports, in the large displacement and deformation fields is proposed for computing the dynamic response to wind loads which blow in arbitrary direction. The initial equilibrium, assumed as the static configuration under self weight and mean wind component, is defined by a continuous approach, following an iterative procedure which starts from the configuration under self weight only. The proposed formulation, which accounts for longitudinal inertia forces, allows to spot the circumstances when the simplified small-sag approach, adopting longitudinal mode condensation, becomes too crude. Numerical simulations have been performed empoying the Proper Orthogonal Decomposition to lower the computational effort.

Introduction

Vibration of cables under wind constitutes an open problem with great implication in practical engineering. Cables are in fact widely used as fundamental structural elements in many applications as bridges, transmission lines, marine risers, cable roof structures. Different sag-to-span ratios are encountered in the various realizations; besides supports can be at the same or at different level.

Traditionally, cable dynamics is intended as the time dependent response around the initial static configuration given by the equilibrium under self weight and described by the catenary equation, including or not cable extensibility or, for shallow cables, by the parabolic approximation [START_REF] Irvine | Cable structures[END_REF] [START_REF] Impollonia | Statics of Elastic Cables under 3D Point Forces[END_REF].

The former needs the solution of a nonlinear equation in order to find cable tension whereas the static deformed shape is written by means of an expression involving hyperbolic functions.

Parabolic assumption does not apply to cables with sag-to-span ratio greater than 1:8, and may lead to crude approximation for inclined cables [START_REF] Irvine | Cable structures[END_REF].

The time dependent response is usually determined with reference to the initial static equilibrium following the work of Irvine and Caughey [START_REF] Irvine | The linear theory of free vibration of a suspended cable[END_REF] in which continuous linear modes of the elastic cable are derived for the in-plane and out-of-plane vibration. While cable elasticity has been shown to be essential in the dynamic of low-sagged cables, the along cable inertia forces are usually neglected by statically condensing the longitudinal equation of motion [START_REF] Luongo | Planar nonlinear free vibration of an elastic cable[END_REF]- [6]. However, the noncondensed approach is more appropriate for describing motion of inclined cables, which exhibit peculiar dynamic behaviour as testified by the veering in frequency curves and the associated hybrid mode shapes [7].

Although a lot of works pay attention to linear and nonlinear cable dynamics, less effort has been devoted to study wind exposed cables. Most of the works about wind-excited cables are related to stay cables or transmission lines and are focused to instability phenomena, such as rainwind or dry-state galloping induced vibration, vortex shedding, galloping of ice-covered cables [8]- [11]. Different papers deal with vibration mitigation by means of external devices such as viscous dampers [START_REF] Impollonia | Dynamic Behaviour of Stay-Cables with Rotational Dampers[END_REF] [START_REF] Main | Evaluation of viscous dampers for stay-cable vibration mitigation[END_REF]. Besides, several experimental works in wind tunnels are devoted to all these problems [START_REF] Zuo | Wind tunnel testing of yawed and inclined circular cylinders in the context of field observations of stay-cable vibrations[END_REF]- [START_REF] Loredo-Souza | A novel approach for wind tunnel modelling of transmission lines[END_REF].

Recently some papers addressed nonlinear cable analysis under turbulent wind. Di Paola et al. [START_REF] Di Paola | Monte Carlo simulation for the response analysis of longspan suspended cables under wind loads[END_REF] use a continuous model for shallow cables assuming a static initial shape based on the parabolic approximation under self weight. The wind is considered to blow perpendicular with respect to the vertical plane containing the cable and Montecarlo simulation is adopted to evaluate cable stochastic response. The case of a cable network under wind loading is considered in [START_REF] Lazzari | Non-linear dynamic analysis of cable-suspended structures subjected to wind actions[END_REF].

Gattulli et al. [START_REF] Gattulli | Dynamics of suspended cables under turbulence loading: Reduced models of wind field and mechanical system[END_REF] attack the problem by a continuous and a FE approach simulating wind time histories through Shinozuka method [START_REF] Shinozuka | Simulation of multivariate and multidimensional random processes[END_REF]; a linear elastic cable behaviour was assumed in the large displacement and small deformation field, applying an updated Lagrangian approach. When the continuous approach is pursued, the expansion of the turbulence components as a superposition of structural modes leads to a reduced wind model which may be viewed as a deterministic version of the well known Proper Orthogonal Decomposition (POD), used in order to determine a reduced stochastic wind field [START_REF] Di Paola | Digital generation of multivariate wind field processes[END_REF]- [START_REF] Chen | Proper Orthogonal Decomposition-Based Modeling, Analysis, and Simulation of Dynamic Wind Load Effects on Structures[END_REF]. The initial shape of the cable is defined by considering self weight and mean wind; thus, the cable lays in a rotated plane with respect to the vertical one, and the layout is defined by means of the parabolic or nonlinear catenary equation according to displacement amplitude. However, the wind is supposed to blow horizontally in the orthogonal direction with respect to cable plane and longitudinal inertia is not accounted for.

A recent paper by Carassale and Piccardo [START_REF] Carassale | Nonlinear discrete models for the stochastic analysis of cables in turbulent wind[END_REF], proposes a continuous model to evaluate the response of shallow cables to turbulent wind around the initial configuration due to self weight. The model adopts a refined wind forces characterization; drag forces act in the plane of the cross sections of the cable and the cable-wind relative velocity is expanded in series up to the third order, introducing nonlinear wind forces contributions. The authors show that the probability distribution of cable response under wind, calculated by means of Montecarlo simulation, can be significantly non-Gaussian.

Purpose of the paper

The formulation proposed herein accounts for longitudinal inertia forces and considers the exact initial equilibrium according to the elastic catenary equation. In this way, the cable model is sufficiently refined to include different scenarios.

The wind is supposed to act horizontally in any given direction, its tangential component is neglected and only the wind component laying in the plane of the cross section is considered as drag force. The initial equilibrium of the cable under self weight and constant wind is reproduced by means of an iterative procedure, starting from the configuration due to self weight only. A finite element model is implemented to evaluate cable dynamics under turbulent wind adopting a modal transformation which retains only a chosen number of modes so to lower the computational effort.

An expansion of drag forces is adopted following [START_REF] Carassale | Nonlinear discrete models for the stochastic analysis of cables in turbulent wind[END_REF] and [START_REF] Macdonald | A unified approach to aerodynamic damping and drag/lift instabilities, and its application to dry inclined cable galloping[END_REF] and the aerodynamic damping matrix is added to structural damping. Wind velocity and cable cross-section are such as to operate in the sub-critical range. The numerical time domain solution is performed in the reduced modal space by simulating coupled modal forces, making use of the POD. So operating the multivariatemulticorrelated process can be transformed into a superposition of multivariate-monocorrelated processes.

The proposed procedure for the nonlinear wind response evaluation of an arbitrarily sagged cable, eventually inclined with supports at different height, can be summarized by the following steps:

• Define mean wind component and direction;

• Determine the initial equilibrium configuration of the cable under mean wind by an iterative procedure;

• Define the finite element model for dynamic analysis under turbulent wind;

• Assign the turbulent wind Power Spectral Density, and determine the cable modal forces with reference to initial equilibrium configuration;

• Evaluate the nonlinear response by using a reduced modal model. Such operations will be discussed in the following sections and applied to different example problems regarding low-sagged, sagged and inclined cables. The Appendix summarizes the order in which the Equations given in sections 3-6 should be applied.

Wind forces on the cable

The cable configurations which will be considered are the followings: the static equilibrium under The distributed drag force on the cable can be defined through a three dimensional vector function as follows:

[ ]

0 0 0 0 0 1 ( , ) ( , ) ( , ) ( , ) ( , ) 2 D D s t c b s t s t s t s t ρ = - - f w z w z ( 1
)
where ⋅ is the norm operator; ρ is air density; D c is the drag coefficient; b is the diameter of the cross section of the cable; 0 ( , ) s t w is the effective wind velocity vector, that is the projection of the wind velocity vector on the cable cross section at abscissa 0 s ; Making the hypothesis of turbulence in the direction of mean wind only and considering the undisturbed wind velocity vector 0 0 0 ( ( ), ) ( , ) ( , ) I s t s t v s t = = v x v k , with constant direction defined by the versor k , the effective wind velocity is written, with reference to configuration II C (see Fig. 2), as follows:

0 0 0 ( , ) (
0 0 0 ( ) ( ) ( ) T s s s = - Γ I t t ( 3
)
where I is the identity matrix of order three and 0 ( )

/ II II s d ds = t
x is the tangent versor to the cable with reference to configuration II C (see Fig. 2), the quantities in Eq. ( 2) become:

0 0 0 1 ( ) ( ) ( ) s s s γ = j Γ k (4) 0 0 0 ( , ) ( , ) ( ) w s t v s t s γ = (5)
In the previous equations, 0 ( ) s γ is defined as follows:

0 0 ( ) ( ) s s γ = Γ k (6)
If the wind velocity is split into mean and turbulent components:

0 0 0 ( , ) ( ) ( , ) v s t v s v s t = + (7)
Eq. ( 2) can be rewritten as:

0 0 0 ( , ) ( ) ( , ) s t s s t = + w w w (8)
where:

0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( , ) ( ) ( , ) ( ) ( ) s w s s v s s s s t w s t s v s t s s γ γ = = = = w j j w j j (9)
According to the quasi-static approach, we assume the hypothesis w z , and expand the drag force in Taylor series with respect to z , following [START_REF] Carassale | Nonlinear discrete models for the stochastic analysis of cables in turbulent wind[END_REF]- [START_REF] Macdonald | A unified approach to aerodynamic damping and drag/lift instabilities, and its application to dry inclined cable galloping[END_REF]. By retaining terms up to the first order one gets:

0 ( ) ( ) ( , ) ... T T D D D s t = = ⎡ ⎤ = + ∇ + ⎣ ⎦ z 0 z z 0 f f f z (10)
where:

1 3 T z z ⎡ ⎤ ∂ ∂ ∇ = ⎢ ⎥ ∂ ∂ ⎣ ⎦ z … (11)
The first term in the expansion:

2 ( ) 0 0 1 ( , ) ( ) 2 D D c bw s t s ρ = = z 0 f j (12)
by substituting the mean and turbulent parts, can be rewritten as follows:

2 2 ( ) 0 0 0 0 0 0 1 ( , ) [ ( ) 2 ( ) ( , ) ( , )] ( ) 2 D D
s t c bw s w s w s t w s t s

ρ = = + + z 0 f j (13)
Furthermore, as the along-wind turbulence only is considered and

0 0 ( ) ( , ) w s w s t t > ∀ , neglecting
the quadratic component of turbulence, Eq. ( 13) becomes:

( ) 0 ( ) 0 ( ) 0 ( , ) ( ) ( , )
D D D s t s s t = = = = + z 0 z 0 z 0 f f f (14)
where:

2 2 2 ( ) 0 0 0 0 0 0 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 D D D s c bw s s c bv s s s ρ ρ γ = = = z 0 f j j (15) 2 (
)

0 0 0 0 0 0 0 0 ( , ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( ) D D D s t c bw s w s t s c bv s v s t s s ρ ρ γ = = = z 0 f j j (16)
The second term in Eq.( 10) can be written as follows:

0 0 0 0 0 0 ( ) 1 1 ( ) ( ) ( ) ( , ) ( ) ( ) 2 2 T T T T D D D c bw s s s c bw s t s s ρ ρ = ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∇ =- + - + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ z z 0 f z I j j z I j j z (17)
The aerodynamic damping force is due to the first term in Eq.( 17), whereas the second term leads to an aeroelastic forcing quantity which is neglected herein. The vector 0 ( , ) s t z is obtained from the cable velocity 0 ( , ) s t u by the relationship:

0 0 0 ( , ) ( ) ( , ) s t s s t = z Γ u (18)
Therefore, the aerodynamic damping force can be written as follows:

0 0 0 0 0 0 0 1 ( , ) ( ) ( ) ( ) ( ) ( ) ( , ) 2 T D s t c b v s s s s s s t ρ γ ⎡ ⎤ = - + ⎣ ⎦ ad f Ij j Γ u (19)

Equilibrium under self weight and mean wind

The mean wind force given by Eq. ( 15) is related to the tangent versor to the cable with reference to configuration II C , according to Eqs. ( 3), ( 4) and ( 6). Such a configuration, i.e. the equilibrium under self weight and mean wind, may be detected by an iterative procedure here conducted for the case of discretized wind forces. Thus, the mean wind force is applied as equally spaced point loads at abscissas 0,i s , each of which is the resultant on a defined interval 0 s Δ ; the i-th point load with 1, 2,..., p i n = can be written as follows:

2 ( ) 0, 0 0, 0 0, 1 ( ) ( ) ( ) 2 i D i D i i s s c bw s s s ρ = = Δ = Δ z 0 P f j (20)
The classic catenary approach which finds initial static shape of the cable under self weight [START_REF] Irvine | Cable structures[END_REF] is extended to allow point loads, generally oriented in space [START_REF] Impollonia | Statics of Elastic Cables under 3D Point Forces[END_REF]. The equilibrium equation becomes:

0 0 0 0 , ( ) ( )
II i i i II d T s s U s s ds = - - - ∑ x R p P (21)
where 0 ( ) T s is the cable tension;

1 2 3 [ , , ] x x x R R R = R
is the vector collecting the three components of the cable tension at the first cable end; Eq. ( 21) gives:

0 0 0 , 0 0 1 1 ( ) ( ) ( ) s II i i i s s U s s ds EA T s ⎛ ⎞ ⎛ ⎞ = -- - + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∑ ∫ x R p P ( 22
)
The vector R can be derived solving numerically the nonlinear integral equation 0 ( )

II L L = x x , i.e.
by imposing the end coordinates of the cable. The first iteration considers distributed forces corresponding to self weight only ( i = P 0) and leads to the vertical configuration I C . The mean effective wind velocity is evaluated as: ( )

0, 0, 0, 0, ( ) ( ) ( ) ( ) i i i i s v s s s γ = w j (23)
0 0 0 0 ( ) ( ) / 1 s II s s T s EA ds = + ∫ (24) where 0, ( ) ( )
i i i T s s U s s = -- - ∑ R p P .
The undisturbed mean wind is defined according to the logarithmic law [START_REF] Simiu | Wind Effects on Structures. Fundamentals and Applications to Design[END_REF]: 0, * 2 0, ( ) ( ) log

I i i r H x s u v s l κ - ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ( 25
)
in which H is the height above ground level of the origin, * u is the shear velocity, κ the von Karman coefficient, r l is the roughness length. Assuming that the wind blows horizontally, intersecting the 1 x axis with an angle ϕ , then [cos 0 sin ] T ϕ ϕ = k .

FEM approach

In order to simplify the finite element formulation, the response variable will be referred to the updated lagrangian coordinate II s which is related to 0 s through Eq. ( 24). This choice allows a more straightforward element definition and the use of constant length elements. The equation of motion of the cable can be written in the following vector form [START_REF] Saitta | Models and Applications for Statics and Dynamics of Cables[END_REF]:

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ ⎢ ⎥ ⎜ ⎟ + + + + = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ x x u u u u f u (26)
where ( , )

II
The approximations , ,

II III II III II III s s m m A A ≡ ≡ ≡
will be adopted and the subscript II will be omitted for the sake of simplicity. In order to derive a finite element solution, the displacement vector is expanded over the element e as follows:

( , ) ( ) ( )

e e s t s t = u N U (28)
where ( ) s N is the shape functions matrix defined as follows:

[ ]

1 2 ( ) ( ) ( ) s s s ψ ψ = N I I (29) with:
( ) The element mass matrix is given by: ( ) ( )

( ) /

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂ ∂ ⎪ ⎪ ⎛ ⎞ - + + + ⎢ ⎥ ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎢ ⎥ ⎪ ⎪ ⎝ ⎠ ⎣ ⎦ ⎩ ⎭ ⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂ ∂ ⎛ ⎞ + + + + - + = ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ∫ ∫ ∫ u u x u x u N u u x u x u B Nf Nu0 (
B A s e T s m s s ds = ∫ M N N (33)
The element stiffness matrix

⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ ∫ x x K BB B B ( 34
) 2 2 1 ( ) ( ) ( ) ( ) 2
⎡ = + + ⎢ ⎣ ⎤ + ⎥ ⎦ ∫ x x G U BU B B BU U B B BU BU (35)
The integral in Eq. ( 35) is solved as:

( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 2
⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = + + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ - + + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ x x Q 0 I BU U B BU BU x x I 0 BU U B BU BU (39)
Through standard assemblage procedures and imposing the boundary conditions of pinned ends at 0 s = and s L = , the global equation of undamped motion is derived as:

( ) ( ) ( )= ( ) t t t + + MU KU G U F (40)
where ( ) t U is the vector collecting the nodal displacements with reference to configuration II C .

A structural damping ratio ζ is considered for all the linear modes; thus, the structural damping matrix is obtained as:

1 T st - - = C Φ ΞΦ (41)
where Ξ is a diagonal matrix with elements 2 i ζω , i ω are natural circular frequencies and Φ is the modal matrix. The aerodynamic damping given by Eq. ( 19) leads to a distributed force ( , ) s t ad f , which can be rewritten as follows:

( , ) ( ) ( , )

s t s st = - ad f D u (42)
where:

1 ( ) ( ) ( ) ( ) ( ) ( ) 2 T D s c bv s s s s s ρ γ ⎡ ⎤ = - + ⎣ ⎦ D I j j Γ (43)
Thus, by Eq.( 28) and Eq. ( 38), the elemental aerodynamic damping force is:

( ) ( ) ( ) ( ) ( ) ( ) B A s e T e e e ad ad s t s s sd s t t ⎛ ⎞ = - = - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∫ F N D N U C U (44)
By standard assemblage procedures:

( ) ( )

ad ad t t = - F C U (45)
where ad C is the assembled aerodynamic damping matrix. The total damping matrix is:

st a d = + C C C (46)
and the damped equation of motion is finally written as:

( ) ( ) ( ) ( ) ( ) t t t t + + = MU + CU KU G U F (47)
The nodal turbulent wind force ( ) t F is assembled starting from the turbulent wind force ( , ) s t f according to Eq. (37).

Introducing the coordinate transformation ( ) ( )

t t = U
Φq , by means of standard modal analysis, the modal equation of motion is derived:

2 ( ) ( ) ( ) [ ( )] ( ) t t t t t + + = q +Ξq Ω q H q Z (48)
where, imposing T = Φ MΦ I , the following quantities are defined:

T = Ξ Φ CΦ , 2 T = Ω Φ KΦ , [ ( )] [ ( )] T t t = H q Φ G Φq , T ( ) ( ) t t = Z Φ F .

Turbulent wind model

The complete definition of wind action needs the characterization of the turbulent stochastic process ( , ) v s t , which is supposed horizontally directed with arbitrary angle with the 1

x axis. The process can be considered a zero mean Gaussian stationary one described in the frequency domain by the Cross Power Spectral Density (CPSD) function. The characterization of CPSD function between different points in space is performed by the definition of the coherence function. The following expression is assumed as coherence function:

( ) ( ) [ ( ), ( ), ] exp ( ) ( ) s i j i j i j i j v v C s s Coh s s v s v s ω ω π ⎛ ⎞ ⋅ ⋅ - ⎜ ⎟ = - ⎜ ⎟ ⎡ ⎤ + ⎣ ⎦ ⎝ ⎠ x x x x ( 49
)
where s C is the exponential decay coefficient, ω is the circular frequency, ( )

i j i i j j i j i j i j v v v v v v v v S s s S S Coh s s ω ω ω ω = x x x x (50) in which ( , ) i i i v v
S s ω is the one-sided PSD of turbulent component of wind velocity. In this paper the following PSD has been employed [START_REF] Solari | Probabilistic 3-D turbulence modeling for gust buffeting of structures[END_REF]:

2 5/3 ( ) ( ) 1 ( , ) ( ) 4
( ) 1 1.5 2 ( )

j j j j j j j j v v v v v L h v s S s I h L h v s δ ω π ω δ π = ⎡ ⎤ + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (51)
where 6.868

δ = , ( )
( ) 300 / 300 v L h h ε =
, ε depends on the ground roughness, ( ) 1/ ln( / )

r v I h h l = ,
and r l is the roughness length. Said H the height above ground level of the origin, 2 ( )

j j h H x s = - .
For each cable nodal point, at abscissa j s ,a three-dimensional matrix ( , ) ( , ) ( , )

j j j j T j j v v s S s ω ω = v v S k k (52)
Projecting on the plane of the cross section: give a diagonal matrix as in the classical case, because out-of-diagonal terms may not be negligible implying that modal forces are not uncoupled in stochastic sense. Finally Eq. ( 57) can be rewritten as follows:

( ) ( ) ( ) ( ) T ω ω ω ω = ZZ S Θ Λ Θ (58)
The numerical time domain solution is then applied to the cable model. The POD expansion Eq.

(58) allows to lower the computational effort of numerical simulation. In fact, the multivariatemulticorrelated process becomes a superposition of multivariate-monocorrelated ones. The PSD of modal forces is used to simulate time histories:

1 ( ) ( ) n j j t t = = ∑ Z r (59)
in which n is the number of modes considered in the analysis; ( ) j t r is a vector of monovariate processes defined as [START_REF] Muscolino | Dinamica delle strutture[END_REF]:

1 ( ) 2 ( ) ( ) cos( ) m j j k j k j k k k t t ω ω ω η ω = = Λ Δ ∑ r Θ (60)
where m represents the number of subdivisions in frequency; ( )

j k ω Λ is the j-th element of the diagonal matrix ( ) ω Λ ; ( ) j k ω Θ
is the j-th column vector of the matrix ( ) ω Θ ; jk η are normal distributed complex random numbers with zero mean and unitary standard deviation. The time integration in the modal space, Eq.( 48), can be performed by means of Runge-Kutta method.

Applications

Four cable layouts, depicted in Figure 3, have been considered in the applications. The first two are related to cables with supports at the same level, the others regard cables with different level of the supports. The cable section and mass per unit length are the same for all the examples and are representative of an ACSR cable for high voltage conductors, which is very sensitive to wind turbulence [START_REF] Karabay | An approach for analysis in refurbishment of existing conventional HV-ACSR transmission lines with AAAC[END_REF]. The geometrical and mechanical properties of the cables are resumed in Table 1.

All the cables are characterized by Irvine's parameters 2 λ falling over the first crossover point.

Cable 1 has the smallest sag, Cable 2 is characterized by a larger value of sag-to-span ratio; both cables have leveled supports (horizontal cables). Cable 3 and Cable 4 are inclined with an angle 30 ϑ = ° (see Fig. 3). The first three cables are subjected to a wind load with / 2 ϕ π In the case of inclined cables, no symmetry or anti-symmetry is encountered. As the velocities determine pressure loads on the cable, this aspect outlines a different loading with respect to classic approaches. The latter are based on continuous modal analysis, where modal shapes and frequencies are evaluated in closed form with reference to equilibrium configuration I C . The refined mean wind load derived by the proposed approach can differ significantly in the case of sagged or inclined cables. On the other hand, for very taut horizontal cables the classic approach is appropriate. In addition, the components of mean wind forces show that for horizontal cables the configuration II C is planar, for inclined cables this is not the case. Thus, this situation can be only roughly described by the classical models.

The comparison of modes with those derived by a classical small-sag formulation which neglects longitudinal inertia forces and condenses longitudinal modes, highlights the differences with the proposed procedure.

The first five modal shapes are depicted in Figure 5, in the global reference system. As the sag increases, the longitudinal component (continuous line) of some modes becomes relevant, whereas in inclined cables is is always significant. It is worth noting that the two inclined cables, which differ between each others only for the wind direction, possess different modes and frequencies; these differences becomes more and more relevant for higher order modes.

In order to compare the modes with those derived by the classical approach, these are depicted in Figure 6, for the horizontal cables, in a rotated reference system, with axes 1 ξ and 2 ξ laying on the plane assumed by the cable under mean wind. The figure shows that the first five modes of Cable 1, the taut one, are very close to those pertinent to a classical analysis on the other hand some differences appear for Cable 2, the sagged one. Table 2 lists for Cable 2 frequencies obtained by static condensation of the longitudinal degree of freedom and those given by the proposed method. The comparison evidences that for large sagged and inclined cables the classic analysis may be inappropriate.

Nonlinear dynamic analysis has been performed by making use of the Runge-Kutta method, numerically solving Eq. ( 48). The modal forces ( ) t Z have been simulated by means of Eq. ( 59) and (60), using the full PSD matrix defined in Eq. ( 58). The projection on the 2 3 ( , )

x x plane, reported in Finally the 0.05 and 0.95 fractiles of the tension along the cable are plotted in Figure 11 and compared with the static value at configuration II C . The 0.05 fractile is smaller than the static value only for Cable 1, whereas in the other cases it results to be larger. It is interesting to emphasize that the dynamic effect on cable tension is much more pronounced when sag increases (Cables 2-4). In the inclined cables, a predictable asymmetry is evident.

Conclusions

The non-linear dynamic response of a cable under turbulent wind has been considered. The dynamic equilibrium is evaluated with reference to the actual initial equilibrium configuration; that is, the static configuration under self weight and mean wind which is attained by an iterative procedure starting from the static equilibrium configuration under self weight only. The proposed procedure allows arbitrary wind direction and accounts for longitudinal inertia forces so to model general problems such as arbitrarily sagged cables and cables with supports at different height. The initial equilibrium configuration is reproduced by a continuous approach based on the catenary equation, whereas non linear motion is described by a non-linear finite element approach. The cable element is defined allowing large strains and displacements, the analysis is performed in a reduced modal space by simulation of wind modal forces adopting a POD expansion.

Applications to four different cable problems show the versatility of the method and its ability to detect specific behaviour. In particular, strong non Gaussianity has been traced for some displacement components and a remarkable increase of cable tension due to turbulent wind is encountered for sagged cables, emphasizing the risk of fatigue failures in suspended cables.

Further investigations, aimed at a deeper comprehension of dynamics of wind-excited cables, should consider the three dimensionality of turbulent phenomenon and represent a future task for the authors.

Appendix

The following steps are required by the proposed formulation 1. First of all define the deformed shape of the cable under self weight 0 ( ) I s x , by making use of the classic catenary approach, so to describe configuration I C . 5. Numerically solving Eq. ( 22) a new static configuration of the cable is obtained and, by Eqs.

(3) and ( 4), the related versor 0 ( ) s j .

Research Highlights

A non-linear FE model of inclined cables is proposed for dynamic response under wind.

The procedure allows arbitrary wind direction and accounts for longitudinal inertia forces.

A remarkable increase of cable tension due to turbulent wind is evidenced for sagged cables.

  plane. It will be assumed that the drag coefficient D c does not depend from Reynold's number (Re), because the flow is considered to be in the sub-critical range (5Re 3.5 10 < ) so that dry-state galloping or rain-induced instability phenomena are ruled out.

  the unitary step function. By using the chain rule of differentiation,

  is the length of the element. Pre-multiplying the dynamic equilibrium equation for the shape function matrix and by means of integration by parts over the element, between abscissas A s and B s , the weak form is obtained:

  31) where ( ) / d s ds = B N is a constant matrix. By introducing Eq. (28), the non linear finite element equation is obtained:

  points j and k of the finite element mesh. Thus, the continuous PSD function is transformed into a discrete matrix ( )

Λ

  is divided into N finite elements, the internal nodal points are N-1 and the global matrix ( )ω ww Swith dimension 3(N-1) can be obtained by assemblage of contributes given by Eq. (53).The POD is resorted to reduce computational effort in numerical simulation, choosing the following PSD matrix[START_REF] Chen | Proper Orthogonal Decomposition-Based Modeling, Analysis, and Simulation of Dynamic Wind Load Effects on Structures[END_REF]: of the PSD matrix have been introduced. The star denotes the transpose of the complex conjugate matrix; indeed, in a general case ( ) could be complex; however, in our case the PSD matrix is a symmetric and real one. Say X the diagonal matrix with elements ( ) of wind forces can be defined through the following matrix equation:By substitution of Eq. (54) in Eq.(55): , instead of the classical wind modes[21][22], the ones given by columns of

2 Dc

 2 Figures 4 show the components of mean wind velocities along the cables at the equilibrium

Figure 7 ,

 7 Figure 7, for Cable 1 resembles an arc of circumference because it behaves similarly to a rigid body

3 .

 3 Define the undisturbed mean wind by Eq. (25).

4 .

 4 Divide the cable into elements of length 0 s Δ so to evaluate the nodal static wind forces by Eq. (20).

Figure 1 .

 1 Figure 1. Cable configurations.

Figure 2 .

 2 Figure 2. Effective wind velocity vector w .

Figure 3 .

 3 Figure 3. Sketches of the cables analyzed in the applications.

Figure 4 .

 4 Figure 4. Mean wind components at static equilibrium II C .

Figure 5 .

 5 Figure 5. First five modes and frequencies of the cables: 1 x component by continuous line, 2 x

Figure 6 . 1 ξ by continuous line, 2 ξ

 612 Figure 6. First five modes of Cables 1 and 2 in a rotated frame, with axes 1 ξ and 2 ξ laying on the

Figure 7 .

 7 Figure 7. Projection on the 2 3 ( , ) x x plane of displacement at / 2 s L = .

Figure 8 .

 8 Figure 8. Probability density function of displacement along 1 x at / 2 s L = (symbols) and Gaussian

Figure 9 .

 9 Figure 9. Probability density function of displacement along 2 x at / 2 s L = (symbols) and Gaussian

Figure 10 .

 10 Figure 10. Probability density function of displacement along 3 x at / 2 s L = (symbols) and

Figure 11 .

 11 Figure 11. Tension along the cable: static value under self weight and mean wind (continuous line);

Table Captions Table 1 .

 Captions1 Properties of the cables.

Table 2 .

 2 Circular frequencies of Cable 2

Table 3 .

 3 Response statistics at s=L/2 for

5 2.25 10 sampling points.

Table 1

 1

	Cable	b	E	A	m	d	H		d	V		2	0 L			s
		() m	2 ( / ) Nm	2 () m	( / ) Kg m	() m	() rad	() m			() m	() rad
	1	0.0308	8 10 10	0.000559	1.73	200		0	0.00	69.88 200.2		/2	0.005
	2	0.0308	8 10 10	0.000559	1.73	200		0	0.00	10493.4 208.0		/2	0.005
	3	0.0308	8 10 10	0.000559	1.73	200		/6	115.47 2246.91 235.0		/2	0.005
	4	0.0308	8 10 10	0.000559	1.73	200		/6	115.47 3387.22 235.0		/4	0.005

Table 1

 1

	Non condensed	Condensed
	0.692	0.692
	1.386	1.386
	1.521	1.386
	2.081	2.021
	2.294	2.081
	2.779	2.778
	3.196	2.778
	3.478	3.441
	3.976	3.478
	4.182	4.182

Table 2

 2

Table 3

 3 CableStandard deviation (m)


Bernard Ycart
email: bernard.ycart@imag.fr

Quantitative Linguistics, and Statistics

Argument

Counting letters in written texts is a very ancient practice. It has accompanied the development of Cryptology, Quantitative Linguistics, and Statistics. In Cryptology, counting frequencies of the different characters in an encrypted message is the basis of the so called frequency analysis method. In Quantitative Linguistics, the proportion of vowels to consonants in different languages was studied long before authorship attribution. In Statistics, the alternation vowel-consonants was the only example that Markov ever gave of his theory of chained events. A short history of letter counting is presented. The three domains, Cryptology, Quantitative Linguistics, and Statistics, are then examined, focusing on the interactions with the other two fields through letter counting. As a conclusion, the eclectism of past centuries scholars, their background in humanities, and their familiarity with cryptograms, are identified as contributing factors to the mutual enrichment process which is described here.

Introduction

In his founding "Manuscript on Deciphering Cryptographic Messages", Ya'kūb ibn Isḥāq Al Kindī (ca. 801-873) wrote the following (Mrayati et al. 2002, 122):

The quantitative expedients include determining the most frequently occurring letters in the language in which cryptograms are to be cryptanalysed. If vowels functioned as the material from which any language is made, and non-vowels functioned as the shape of any language, and since many shapes can be made from the same material, then the number of vowels in any language would be greater than non-vowels. For instance, gold is the material of many shapes of finery and vessels; it may cover crowns, bangles, cups, etc. The gold in these realizations is more than the shape made of it. Similarly, the vowels which are the material of any kind of text are more than the non-vowels in any language.

Six centuries later, Leon Battista Alberti (1404Alberti ( -1472) ) echoed (Alberti 2010, 173): I could go on at great length about this, but as briefly as possible, just let me say that the arrangement and the number of letters constitute various syllables which, when combined, form finished words, which differ in sound and meaning. First I shall speak of number and how this is connected to numeric ratios. This phenomenon above all involves the vowels. We will begin with them. A syllable is constituted of either a single vowel, a consonant associated to a vowel, or several consonants united to a vowel. Without vowels there are no syllables. Thus if we take, for example, one or two pages of poetry or prose and extract the vowels and consonants, listing them in separate series, vowels on one side and consonants on the other, you will no doubt find that there are numerous vowels.

There is no evidence that Alberti had any knowledge of earlier Arab cryptology treatises, which were rediscovered only recently [START_REF] Kadit | Origins of Cryptology: the Arab contributions[END_REF]. Actually, many similar observations can be found in Cryptology treatises along the centuries. All have in common a very prosaic, long time occupation of scholars: counting letters. As Georges Udny Yule (1871Yule ( -1951) ) said: "It is full of interest, and is of the most venerable antiquity and august associations[…]" (Yule 1944, 7).

Since Al Kindī, all Cryptology treatises have included a section on monoalphabetic simple substitution cyphers, which consist in replacing each letter by a given character, through a one-toone correspondence. If the language of the encrypted message, and the frequency order of the letters in that language are known, the code is vulnerable to so called frequency analysis: the most frequent character in the message stands for the most frequent letter of the alphabet, the next more frequent for the second most frequent letter, and so on. This seemingly straightforward procedure conceals a major difficulty, which took a very long time to understand: there is no such thing as the frequency of letters. Along the centuries, many scholars have struggled with that difficulty, gradually developing notions that went far beyond technical necessities of cryptology, and eventually helped shaping two other fields: Quantitative Linguistics and Statistics. Throughout their history, the three disciplines have constantly nourished each other, and letter counting has been the catalyst of that mutual enrichment process: its history is the object of the present article.

The historical development of each of the three disciplines is well documented. For Cryptology, basic references are [START_REF] Kahn | The codebreakers -The Story of Secret Writing[END_REF] and [START_REF] Mollin | Codes: the guide to secrecy from ancient to modern times[END_REF]. Here, our focus will be on the perception of letter frequencies among the authors of Cryptology treatises, from Al Kindī to 20 th century textbooks. Arab authors often gave tables of letter frequencies, based on counts of several thousand letters, and they appeared quite aware of the necessity of a large sample size. Western treatises after Alberti contrast: quantified indications remain scarce, and large scale counts do not appear before the 19 th century. Alberti's "De Componendis Cifris" is a quite singular point in that history: it went notably further than cryptology, giving the first example of a quantified stylistic difference ever. After Alberti, the emergence of Quantitative Linguistics was slow. Before the 19 th century, it can only be traced through isolated remarks. In the second half of the 19 th century, authorship disputes, in particular the Bacon-Shakespeare controversy, fueled more systematic research. A detailed history of quantitative authorship attribution appears in [START_REF] Grieve | Quantitative authorship attribution: a history and an evaluation of techniques[END_REF]. We shall complete it, presenting earlier sources on the proportion of vowels to consonants.

On the history of Statistics, we shall refer to [START_REF] Hacking | The Taming of Chance[END_REF], and [START_REF] Stigler | Statistics on the table: the history of statistical concepts and methods[END_REF]. A major change took place in the perception of statistical concepts, during the second half of the 19 th century: partly inspired by Darwin's theory of evolution and the controversy around Quetelet's "Average Man", statistics gradually moved from the mechanist perception inherited from astronomy, of a ʿtrue valueʾ measured with random errors, to the modern notion of intrinsic variability. Letter frequency distributions were regarded by early statisticians, in particular Babbage and Quetelet, as one of the "Constants of Nature and Arts", just as they had always been viewed by cryptologists and linguists. The perception gradually evolved to the modern notion of an information source defined as a stochastic process, through the works of Markov, then Kullback and Shannon: Cryptanalysis was crucial in the creation of information theory. At the same period, Zipf's observations on the relative frequency of words initiated a seemingly different corpus of studies in Statistics. Both converged in the second half of the 20 th century through the works of Mandelbrot [START_REF] Brillouin | Science and information theory[END_REF].

A short history of letter counting will first be presented. Then the three domains, Cryptology, Quantitative Linguistics, and Statistics, will be examined in that order, focusing on the interactions with the other two fields through letter counting. As a conclusion, the eclectism of past centuries scholars, a generalized background in humanities, and the familiarity with cryptograms, will be identified as contributing factors to the mutual enrichment process which is described here.

Letter counting: from rope-dancing to industrial mechanics

When Yule referred to text counting as being "of the most venerable antiquity and august associations", he meant the Jewish tradition of the Masorah, about which he cited Hasting's "Dictionary of the Bible" (Yule 1944, 7, see also [START_REF] Kelley | The Masorah of Biblia Hebraica Stuttgartensia[END_REF]):

They counted the verses and the words of each of the 24 books and of many sections; they reckoned which was the middle verse and the middle word of each book; nay, they counted the letters both of particular sections and even of whole books. Thus they could specify the middle word and the middle letter in the Torah, the middle verse and the middle letter of the Psalms. They counted also the frequency of occurrence of words, phrases, or forms, both in the whole Bible and in parts of it.

Thus, for proofreading or salary purposes, Bible copyists have counted letters and words for many centuries. Yet, even before the Masoretes, long before any text had ever been counted, evidence was there that letter frequencies could be biased at will. Some authors did not fail to see it. Here is what 'Alī ibn ad-Durayhim (1312-1361) remarked (Mrayati et al. 2003b, 100). This is the letter order of frequency in the Holy Koran, although this order may differ in other language usages. Some deliberately encipher poetry and prose, dispensing with the letter [ā], or without letter-dotting or without idle particles [those that do not affect the parsing of what follows].

Dispensing with one or several letters in writing is the principle of lipograms, already practiced by some Greek poets [START_REF] Perec | Histoire du lipogramme[END_REF]. Georges Perec, being himself the author of an impressive novel of about 360,000 letters that does not contain a single e, had perfectly seen the relation between lipograms, cryptology, and the Holy Scriptures: "the Book is a cryptogram of which the Alphabet is the cipher" (Perec 1973, 73). Since by lipograms or any other form of constrained writing, letter frequencies can be skewed, why then did all cryptologists, linguists, and statisticians fail to mention it before the end of the 19 th century? Part of the explanation may be that word games, even practiced by the best authors, were not taken very seriously before the 20 th century. In his book on anagrams, Henry Benjamin Wheatley devotes 3 pages to the history of lipograms (Wheatley 1862, 21-23).

Here is how they begin.

Lipograms are perhaps the most ridiculous of those follies; it is of them that De Quincey speaks when he says "Recalling that rope-dancing feat of some versewriters, who through each several stanza in its turn, had gloried in dispensing with some one separate consonant, some vowel or some diphtong, and thus achieving a triumph such as crowns with laurel that pedestrian athlete who wins a race by hopping on one leg, or wins it under the inhuman condition of confining both legs within a sack."

After the art of printing was invented, letter counting continued, for technical reasons: typographers needed to know what quantities of each character had to be prepared. In 1828, Adolphe Quetelet (1796Quetelet ( -1874) ) is not yet the founding father of statistics in social sciences. When he publishes in his new journal, an article entitled "On the ratios between the number of letters that compose the alphabet in different languages", he does so under the section "Industrial Mechanics", and not Statistics as one would have expected from him (Quetelet 1828, 339-341). Among his motivations, typography clearly comes before cryptology:

It is important for typographers and foundry workers, to know the proportions of letters that compose the alphabet in the different languages, in order to form their stock. The knowledge of these proportions may also interest the philologist; on the other hand, suspicious governments have sometimes used it to decipher correspondences in which the letters of the alphabet were substituted one for the other. Since these proportions are generally little known, we have thought appropriate to record here those of three alphabets [Dutch, French, Italian].

In the following volume, Quetelet renders a detailed account of a thesis defended by Marcel Hayez, from the family of typographers who publishes the "Correspondance Mathématique et Physique" (Quetelet 1829, 391-394). Hayez has provided detailed letter and character counts for French, Latin, Italian, Portuguese, Greek, English, Dutch, and German, over corpuses of several thousand letters; quite naturally, typography is both his motivation and his application domain.

That typographers might have a precise idea of character frequencies, came to the mind of Samuel Morse in 1838 when he was in search for a binary code that would optimize time, hence match the simplest codes to the most frequent letters. Instead of painstakingly counting letters by himself, he simply walked to a printing office and noted the quantities that he found there (Morse 1914, 69). Both letter frequencies and the Morse alphabet later contributed shaping the QWERTY keyboard, that we are still using [START_REF] Yasuoka | On the prehistory of QWERTY[END_REF]. Contrarily to Morse, it seems that Alfred Butts, the inventor of the Scrabble, counted letter frequencies from newspapers front pages, to decide the value and number of the letters in his new game, at least if web resources are to be believed on that point.

Cryptology: don't spare either labor or paper

According to usage, we distinguish cryptography (ciphering) from its contrary cryptanalysis (deciphering). Not all Cryptology treatises deal with cryptanalysis. Those who do, usually begin with a section on letter frequencies. Some quotations, given in chronological order, will help understand how the frequency analysis method has been perceived by cryptographers along the ages. A very precise and rational procedure for the Arabs, it came to be one of the tools of a largely divinatory art among westerners of the 15 th to 18 th centuries.

The clearest exposition of the method may be the first one, by Al Kindī (Mrayati et al. 2002, 124):

Among the expedients we use in cryptanalysing a cryptogram if the language is already known, is to acquire a fairly long plaintext in that language, and count the number of each of its letters. We mark the most frequent letter ʿfirstʾ, the second most frequent ʿsecondʾ, and the following one ʿthirdʾ, and so forth until we have covered all its letters. Then we go back to the message we want to cryptanalyse, and classify the different symbols, searching for the most frequent symbol of the cryptogram and we regard it as being the same letter we have marked ʿfirstʾ in the plaintext; then we go to the second frequent letter and consider it as being the same letter we have termed ʿsecondʾ, and the following one ʿthirdʾ, and so on until we exhaust all the symbols used in this cryptogram sought for cryptanalysis. Indeed, Al Kindī had based his counts on 7 pages totaling 3667 letters, a feat that westerners were not to accomplish for a very long time. Three centuries later, 'Afīf ibn 'Adlān (1187-1268) proposed a classification of letters into three frequency classes, and introduced frequencies of digrams (Mrayati et al 2003a). His contemporary 'Ibrāhīm ibn Dunaynīr (1187-1229) repeated Al Kindī's counts, and could only confirm: "So it came home to me the validity of the statement of Ya'kūb Al Kindī, peace be on his soul." Here is how he explains the method (Mrayati et al. 2005, 84).

I say: The type of encipherment characterized by changing the forms of letters is achieved by devising shapes or symbols not attributed to letters at all. In this method every letter is represented by a symbol that is unique to it. The cryptanalysis is accomplished by counting the symbols of the cipher message, and establishing the frequency of occurrence for each symbol, by affixing the frequency number to the respective symbols of the cipher. Having done that, you dispose the symbols in order of frequency precedence, designating the most frequently-occurring symbol in its locations throughout the cryptogram. Do the same with the next frequently-occurring symbol, and so forth until you exhaust all the symbols of enciphered letters. Now place the highest-frequency symbol against the highest frequency letter of the Arabic alphabet, doing the same with the rest, conformably with their order of frequency. Keep going in the same vein until you use up all the letters and symbols. Some Arab treatises went further than Al Kindī on particular linguistic or statistical points; but from the point of view of Cryptology, the essential was there. Confronted to the same problem, cryptographers of the Italian Renaissance expressed their findings in a different way. The first of them is Alberti, already cited in the introduction. Apart from a very interesting linguistic remark that will be examined in the following section, Alberti does not give exact frequencies, but rather indications of relative orderings, and mutual relations between letters. After a careful (but not quantified) linguistic examination of Latin, he somewhat maliciously concludes (Alberti 2010, 177):

We have discussed the number and the order of the letters and those vowels that are most frequently encountered in writing, of the possible association of vowels with regards to the consonants, we have said which are most frequently encountered and regarding their arrangement, how they behave in addition or in sequence. From what has been said thus far you will quite easily deduce (if I am not mistaken), how astute minds will perceive a way to decipher and interpret coded writing in a most easy way.

A few years after Alberti, when Cicco Simonetta (1410-1480) exposes his method for deciphering, his first seven rules aim at deciding whether the enciphered text is in Latin or "Volgare." In the other 6 rules, he does give observations to isolate vowels, and recognize the other letters, but never mentions frequencies [START_REF] Perret | Les règles de Cicco Simonetta pour le déchiffrement des écritures secrètes.(4 juillet 1474)[END_REF]. Italian cities, and in particular the Vatican, are known to have had excellent cryptanalysts for a long time; among them the Argenti family. After his dismissal in 1605, Matteo Argenti compiles an impressive manual, exposing his method of cryptanalysis quite clearly. Sixty two rules are listed; only the 22 nd mentions that frequent letters are likely to be vowels (Meister 1906, 166).

At about the same period, François Viète (1540-1603) served as the decipherer of the French court. His notes, written shortly before his death, prove that he used frequency analysis, even though it was only one among several other tools (Pesic 1992, 13):

One must note all the sorts of figures, whether ciphers or jargon, and count how many times they occur, then note all the sorts of figures which precede or which follow and compare the most frequent in order to discover the same words, and the same meanings. Don't spare either labor or paper.

By the end of the 17 th century, the frequency analysis method was well understood by everyone in the trade, and even popularized in treatises, such as John Falconer's (Falconer 1685, 8-10).

To proceed regularly therein, you must endeavour to distinguish betwixt the Vowels and Consonants. And first, the Vowels generally discover themselves by their frequency; for, because they are but few in number, and no word made up without some of them, they must frequently be used in any Writing : however, it may by accident fall out, that some of the Consonants shall be oftener found in an Epistle.

[…] To distinguish one Vowel from another, after you have made the most probable Suppositions in separating them from the Consonants, 1. Compare their frequency, and e, as we observed before, is generally of most use in the English Tongue, next o, then a and i; but u and y are not so frequently used as some of the Consonants.

[…]

To distinguish one Consonant from another, you must 1. (as before) observe their frequency. Those of most use in English are d, h, n, r, s, t, and next to those may be reckoned c, f, g, l, m, w; in a third rank may be placed b, k, p; and lastly, q, x, z.

David Arnold Conradus, published in 1732 a Latin book Cryptographia denudata, sive Ars Decifrandi, [START_REF] Conradus | Cryptographia denudata, sive Ars Decifrandi[END_REF], of which he gave, ten years later, a more widely accessible version in a series of articles published by "The Gentleman's Magazine" (Conradus 1742, 134).

Prop 2] Every Language has, besides the Form of its Characters, something peculiar in the Place, Order, Combination, Frequency, and Number of the Letters.

Rule 1] In Deciphering, Regard is to be had to the place, Order, Combination, Frequency, and number of the Letters […] Prop. 6] The Vowels, generally five, are four times out-numbered by the Consonants, the Vowels must therefore recur most frequently.

Rule 6] The Letters that recur most often are Vowels.

To the best of our knowledge, only one quantified assertion about frequencies can be found in a Cryptology treatise between Alberti and the 19 th century: it appears in Christian Breithaupt's "Ars Decifratoria" (Breithaupt 1737, 109); it is much less precise than Alberti's observations. The frequency of letters should be noted in general, since in any language vowels are more numerous than consonants. The reason for making these observations is obvious. Actually, for a given number of vowels, the corresponding number of consonants must be larger by five fourths; it cannot be otherwise, vowels being more frequent than consonants.

Half a century after Conradus andBreithaupt, Philip Thicknesse (1719-1792) describes again the same deciphering method, with hardly any novelty (Thicknesse 1772, 22-25).

The next thing to observe is, what letters occur oftenest, and those you may conclude are vowels, and that which is most frequent, to be an e -as e in English occurs oftener than any other letter.

[…] Observe also, that i, in English, never terminates a word, nor a or u except in sea, you; or those; and thus by comparing the frequency of the letters, you will generally find e occur the oftenest: next o, then a, and i; but u and y, are not so often used as some of the consonants. Among the vowels, e and o are often doubled; the rest scarce ever: and e and y, often terminate words, but y is less frequent, and consequently easily distinguished. To find out one consonant from another, you must also observe their frequency, d, h, n, r, s, t, and next to those, c, f, g, l, m, w, in a third rank may be placed b, k, p, and lastly q, x, z. This remark, however, belongs to English; for in Latin the consonants are l, r, s, t; next c, f, m, n; then d, g, h, p, q; and lastly, b, x, z.

Admittedly, before computers, counting thousands of letters to establish a frequency table was a time consuming and not very interesting activity. Most 20 th century manuals contain such tables, but rare are those who do not reproduce earlier counts. For instance, in the French edition [START_REF] Sacco | Manuel de cryptographie[END_REF] of one of the most translated and re-published cryptology manuals, the source of the tables is cited as [START_REF] Valerio | De la cryptographie[END_REF]). We have seen [START_REF] Klüber | Kryptographic lehrbuch der geheimschreibekunst[END_REF]) cited as a source of tables, yet no tables were available in the edition that we could consult. The first European manual of Cryptology to give frequency tables for different languages, that we are aware of, is Charles François Vesin's "Traité d'Obscurigraphie" (Vesin 1838, 100-104). As we have seen, similar tables had already been established earlier, by Quetelet and Hayez. Vesin only announces "having tried a corpus of one thousand letters." The table comes with some warning: "There may doubtlessly be some difference in the distribution of letters, between some period of writing and another period; but this difference will never be exorbitant." By the turn of the 20 th century, such warnings became more systematic. Yet, their authors still firmly believed in the stability of letter frequencies for ʿnormalʾ writing. Author of a "Manual for the solution of military cipher", intended for Army Service Schools shortly before the United States entered WWI, Parker Hitt considers that deciphering offices "should be provided with tables of frequency of the language of the enemy, covering letters and digraphs." Explaining what he means by "tables of frequency", he appears aware of the stylistic problem, which he downplays (Hitt 1916, 4).

In fact, if ten thousand consecutive letters of a text be counted and the frequency of occurrence of each letter be noted, the numbers found will be practically identical with hose obtained from another text of ten thousand letters in the same language. The relative proportion of occurrence of the various letters will also hold approximately for even very short texts.

Such a count of a large number of letters, when it is put in the form of a table, is known as a frequency table. Every language has its own frequency table and, for any language, the frequency table is almost as fixed as the alphabet. There are minor differences in frequency tables prepared from texts on special subjects. For example, if the text be newspaper matter, the frequency table will differ slightly from one prepared from military orders and will also differ slightly from one prepared from telegraph messages. But these differences are very slight as compared with the differences between tables of two different languages.

Hitt's presentation is still quite far from the modern view of frequency tables. Here is how they are described in a present day manual (Bauer 2006, Chapter 15). Moreover, even long texts normally show considerable fluctuations of character frequencies. ʿTheʾ frequency distribution of English is a fiction, and at best the military, diplomatic, commercial, or literary sub-languages show some homogeneity; indeed even the same person may speak a different language on the circumstances. Correspondingly, statistics on letter frequencies in different languages are quite variable. Moreover, most of the older counts were based on texts of only 10,000 or fewer letters. For the frequency ordering there are already great differences in the literature.

After having traced the frequency analysis method through cryptology manuals, several questions remain to be answered. The first one is: why was the method considered useful for such a long time? Many manuals, after having exposed the vulnerability of monoalphabetic substitution, propose alternatives. Alberti had already described a polyalphabetic cipher, soon improved by Tithemius, Bellaso, and Vigenère [START_REF] Kahn | The codebreakers -The Story of Secret Writing[END_REF][START_REF] Mollin | Codes: the guide to secrecy from ancient to modern times[END_REF]. Why weren't these ciphers systematically used from them on? In 1839-1841, Edgar A. Poe earned himself a high reputation as a cryptanalyst, by solving simple substitution ciphers. However, he does not seem to have had much more competence than he had found in [START_REF] Conradus | The art of deciphering[END_REF]): see [START_REF] Friedman | Edgar Allan Poe, Cryptographer[END_REF](Poe &Peithman 1986, 283). According to [START_REF] Kahn | The codebreakers -The Story of Secret Writing[END_REF], codes that were vulnerable to frequency analysis remained in use throughout WWII. The explanation usually retained is that polyalphabetic ciphers were too complicated to use, in particular by military services. Another explanation may be diplomatic secrecy. From the 15 th century on, most courts used their own ciphers, usually more complicated than monoalphabetic substitution, yet still vulnerable to frequency analysis; but no one wanted the others to know that they could solve their codes. This will be illustrated through a well known anecdote about Viète's feats as a cryptanalyst in the diplomatic war between the kings of France and Spain, Henry (Henri IV) and Philip (Felipe II). Here is how it is reported in (Kahn 1996, 82).

Meanwhile, Philip had learned, from his own interceptions of French letters, that Viète had broken a cipher that the Spanish -who apparently knew little about cryptanalysis-had thought unbreakable. It irritated him, and thinking that he would cause trouble for the French at no cost to himself, told the pope that Henry could have read his ciphers only by black magic. But the tactic boomeranged. The pope, cognizant of the ability of his own cryptologist, Giovanni Batista Argenti, and perhaps aware that papal cryptanalysts had themselves solved one of Philip's ciphers 30 years before, did nothing about the Spaniards' complaint; all Philip got for his effort was the ridicule and derision of everyone who heard about it.

The historical context is reported by [START_REF] Pesic | François Viète -father of modern cryptanalysis[END_REF] in similar terms: "Spain seems not no have had cryptanalysts, which might explain Philip's attitude." Complacently spread by many French sources along the centuries, that story resembles more propaganda than historical truth. At the same period, Philip also had his own cryptanalyst, Luis Valle de la Cerda, who was as much able to decipher Henri's dispatches as Viète was with Philip's (Carnicer Garcia & Rivas 2005, 73). Contrarily to Henry, Philip probably preferred the French to have some amusement at his own expense, rather than letting them know about Valle de la Cerda's achievements.

Secrecy may not have been the only reason for the relative lack of popularity of frequency analysis among cryptologists. It must be emphasized that the method was neither necessary nor sufficient to solve the ciphers cryptanalysts were usually confronted with. That it was not necessary has been illustrated in particular by Simonetta and Argenti: there are many other ways to guess vowels, by looking for isolated characters, word endings, digrams, and trigrams. The method was not sufficient for two reasons. One is the length of encrypted messages: many authors have pointed out that the shorter the message, the less reliable the order of frequencies. That point will be examined in the Statistics section. The other reason is that, even when a monoalphabetic code had been chosen, plenty of ways to level out frequency differences were available, and recommended in manuals; the simplest was to alternate different characters for the same letter. Using or not frequency analysis, actual deciphering was an arduous task, requiring at least as much intuition as method; hence Viète's exhortation: "don't spare either labor or paper!"

Quantitative Linguistics: if sholars knew the law of averages as well as mathematicians…

Reading the collection of the "Arabic Origins of Cryptology" (Mrayati el al. 2002 ff.), one cannot but remark how Arab authors often went much further into the analysis of the language, than strictly needed for cryptanalysis technical purposes: their treatises contain many profound remarks on the structure of language, on the alternation of vowels and consonants, on the differences between poetry and prose, on rhythm and prosody. The symbiosis between Arab Cryptology and Language Sciences is described in (Mrayati et al. 2002, 46-53). As stated there, the relationship "was not unilateral. Biographical sources confirm that the leading scholars of language were also involved in cryptology." The same can be said of many western scholars. Our purpose here is not to analyze the relationship between Cryptology and Linguistics in general, but rather to describe, through some particular examples, how some linguistic analyses of the language, involving letter counts, may have fostered the emergence of Quantitative Linguistics as a new field. Historical accounts of Quantitative Linguistics written so far have given the main role to autorship attribution problems (e.g. [START_REF] Williams | Studies in the History of Probability and Statistics: IV. A note on early statistical study of literay style[END_REF][START_REF] Bailey | Statistics and Style: a historical survey[END_REF][START_REF] Peng | Quantitative analysis of literary styles[END_REF]. This can be explained on the one hand by the hotly debated Bacon-Shakespeare question [START_REF] Stopes | The Bacon-Shakspere question[END_REF], on the other hand by the influential work of [START_REF] Yule | The statistical study of literary vocabulary[END_REF]. We believe that the history of the field should be re-evaluated, taking into account earlier discussions on the vowel-consonant proportion, three examples of which are given in this section.

A remark in ad-Durayhim's treatise (Mrayati et al. 2003b, 100), already cited, shows that he was aware of possible differences in letter frequencies, under a change of style. Yet we have not seen any such remark being quantified in Arab treatises. To the best of our knowledge, the first stylometric observation in history is due to Alberti, in "De Componendis Cifris" [START_REF] Alberti | De Componendis Cifris[END_REF], 173, Meister 1906, 127).

From my calculations, it turns out that in the case of poetry, the number of consonants exceeds the number of vowels by no more than an octave, while in the case of prose the consonants do not usually exceed the vowels by a ratio greater than a sesquialtera. If in fact we add up all the vowels on a page, let's say there are three hundred, the overall sum of the consonants will be four hundred.

That remark has been analyzed in [START_REF] Ycart | Alberti's letter counts[END_REF]). The interpretation proposed there is the following. The proportion of vowels for poets may be denoted by P, being aware that what precisely is meant by ʿpoetsʾ and ʿvowelsʾ, is a matter of convention. Alberti's assertion can be mathematically translated into (1-P)-P<1/8, or else P>7/16. Alberti opposes poets to "rhetores", i.e. orators, for which the proportion of vowels against consonants is said to be "sesquitertia" in the Latin text, or else four against three. If R denotes the proportion of vowels for orators (again a matter of convention), what is stated is (1-R)/R < 4/3, or else R>3/7. The difference between 7/16=43.75% and 3/7=42.86% is smaller than 1%. Yet, using a corpus of Latin texts, from poets and orators, it could be proved that there was indeed a significant difference in the use of vowels, and that Alberti could have detected that difference by letter counts. Moreover, the last part of the citation gives an idea on his sample sizes: mentioning a page of 300 vowels and 400 consonants might have been a way of indicating that his observations were supported by counts on large enough sets of letters. Because of that remark, some have seen in Alberti a pioneer of ergodic and information theories (Mercanti & Landra 2007, note 14). This may be exaggerated; notwithstanding, the two sentences quoted above are quite singular. Alberti himself, while commenting at length vowels, consonants, their relative frequencies and positions, never gives any other number. As already stated, the only other quantified remark found in a Cryptology treatise, that of Breithaupt, falls short of Alberti's observation as far as accuracy is concerned.

Our next example does not come from Cryptology, yet it deals with the same problem of vowel-consonant proportion. William Owen Pughe (1759-1835) a grammarian and lexicographer, devoted his life to the illustration and defense of the Welsh language, producing in particular a grammar, and a monumental Welsh-English dictionary. Between 1796 and 1816 at least, he regularly contributed to "The Monthly Magazine", under the pen name of "Meirion", sending letters on etymology, poetry, "the affinity of Welsh and Hebrew", and so on (Davies 2002, 36). On January 1st, 1799, he wrote the following (Pughe 1799).

In looking into the new edition of Chamber's Cyclopaedia, sometime ago, I casually met with a remark upon a subject, which had relation to language, wherein the Welsh and the Dutch were pointed out, as abounding more with consonants, than most, if not all the European tongues. I well knew that such a statement was proverbial, as a vulgar prejudice: but I became a little angry, at finding it had obtained a place in one of the first philosophical dictionaries of the present age; and, not being able to efface the impression from my mind, I had recourse to the finding a tolerably exact arithmetical certainty, as to the fallacy of such an observation. The method, adopted as the most eligible, was to fix upon the mean number of vowels to a hundred consonants, in different languages; and to exhibit the results in a table. As the conclusion, to be drawn from it, tends to establish a point, if not of importance, at least of some curiosity, you may be induced, Sir, to give it insertion, in your valuable repository.

[table for Welsh, Greek, Spanish, Italian, Latin, French, German, Dutch, English] As the French, and the English, differ so considerably, in pronunciation, from what they appear in orthography, the following comparison shews the reduced numbers of the vowels and the consonants[…] The English is very variable, with respect to the proportion of vowels and consonants: that of the consonants is much greater in the scripture style, than in elegant writing, and more especially that which is scientific, from its containing more words derived from the learned languages. In the bible, the compass of the variation, in the number of vowels, is generally from about 68 to 50; but the medium may be settled at 56 to a 100 consonants. In polished writing, the medium number of vowels may be settled at 66; and the mean between the two styles will be 61, the number inserted in the foregoing table .  The compass of variation in the Greek is considerable. I have found 150 vowels to 100 consonants; and frequently as low as 86. The other languages are pretty close to the average number, given in the table; the Welsh seldom deviates three vowels from the mean number.

By modern standards, some scientific objectivity might be wanting. Yet, with Statistics still in its infancy, Pughes shows a quite remarkable care in assessing the variability of his counts and find a "tolerably exact arithmetical certainty": recall that by 1799, the word "Statistics" had just been given its modern meaning by Sir John Sinclair a few years before (Hacking 1990, 26). Counting the number of vowels per one hundred consonants is unusual, but perfectly correct if enough samples are taken to assess the "medium" number of vowels, together with a reasonable "compass of variation." The observation that the ratio vowel-consonant depends not only on the language, but also on the style is there, and also the distinction between pronunciation and orthography. In his defense of Welsh, Pughe means to prove that it has more vowels than the other tongues, and he accepts (and even supports) the idea that "the harmoniousness of a language depends much upon the proportion of the vowels to the consonants." The sentence comes from the article "Vowel" of the Encyclopedia Americana, and expresses what was a commonplace prejudice in the beginnings of comparative linguistics. It was written by Francis Lieber (1800-1872), an early specialist of linguistic relativism (Andresen 1996, 114-119). We shall quote from his unusually long article "Consonants" of the same Encyclopedia Americana [START_REF] Lieber | Consonants" in Encyclopedia Americana[END_REF]. There, he exposes a statistical study of his, comparing the ratio vowel-consonant among classical poets of different languages.

It occurred to the writer, while preparing this article, that it would lead to interesting results, if the proportion of the vowels and consonants, in the different languages, could be ascertained; but the conclusions, to which he has been led by such investigation as he has bestowed on the subject, are rather to be regarded as indications of what might be learned from more thorough inquiries, than as facts from which general deductions can be safely drawn. In making the comparison, passages have been taken from the popular poets of different countries. The different passages were in the same measure, or in measures very similar, so that the number of syllables in each would be very nearly the same.

Of course, Lieber distinguishes the "orthographic proportion" from the "phonic proportion." He also knows that euphony cannot only come from the frequency of vowels. Interestingly, he is quite cautious on methodology issues and aware of the necessity of a rigorously selected corpus of sufficient size.

To give any thing like accuracy to such investigations, it is obvious that the results ought to be taken both from prose and poetry, also from many different writers, and the language of conversation.

In the conclusion of the article, Lieber clearly expresses his ambition: nothing less than a new branch of Linguistics, that would quantify the elements composing a language, exactly as the then newly founded stoechiometry quantifies atoms and molecules in Chemistry: quite an interesting comparison for a German born political philosopher, trained in Mathematics at the University of Jena (see [START_REF] Hufbauer | The formation of the German chemical community (1720-1795[END_REF].

It is easily seen, that, in the languages of Latin origin, the proportion of consonants to vowels is much smaller than in the Teutonic idioms. To compare the proportions of consonants to vowels, in such different families of languages; to show the proportions of the gutturals, labials, &c., of the different idioms; and again, the proportion of these letters in the various families of languages, or according to the different parts of the earth to which they belong, as Asiatic, European, &c. languages, and many other calculations -might lead to very interesting conclusions. This branch of philology might be compared to the new department of stoechiometry in chemistry, which treats the proportions of the quantities of the elements in a state of neutralization or solution -a branch of science which every day becomes more important, and which has been illustrated by the labors, past and present, of a Berzelius, Klaproth, Döbereiner and others.

Lieber's manifesto came 16 years before another declaration of intent, much less precise, yet most often cited as a birth date for Statistical Linguistics [START_REF] Bailey | Statistics and Style: a historical survey[END_REF][START_REF] Grieve | Quantitative authorship attribution: a history and an evaluation of techniques[END_REF]: a letter to one of his friends by the famous mathematician Augustus De Morgan (1806Morgan ( -1871)), dated August 18, 1851, where he suggested statistical counting to settle authorship disputes (De Morgan 1882). De Morgan did not mean to go through the tedious process of counting by himself:

It has always run in my head that a little expenditure of money would settle questions of authorship in this way. The best mode of explaining what I would try will be to put down the results I would expect as if I had tried them.

In the conclusion of his letter, De Morgan expresses his absolute trust in the law of large numbers: he is perfectly acquainted with the statistical developments of his time, but the paradigmatic shift of statistics had not yet taken place [START_REF] Hacking | The Taming of Chance[END_REF]).

If scholars knew the law of averages as well as mathematicians, it would be easy to raise a few hundred pounds to try this experiment on a grand scale. I would have Greek, Latin, and English tried, and I should expect to find that one man writing on two different subjects agrees more nearly with himself than two different men writing on the same subject. Some of these days, spurious writings will be detected by this test.

A complete account of how De Morgan's ideas, published after his death, were implemented by Thomas C. Mendenhall (1841Mendenhall ( -1923)), and eventually developed into a full methodology for quantitative authorship attribution, has been given by (Grieves 2005). It will not be reproduced here. The interested reader is also referred to [START_REF] Yule | The statistical study of literary vocabulary[END_REF], already cited. It is quite instructive to compare the idealistic confidence of De Morgan, who did not actually count letters, and wrote before the great statistical shift, to the wariness of Yule, who was one of the very best statisticians of his time, and had counted himself many texts (Yule 1944, preface).

When I had advanced some way in that particular study, it became only too clear into thorny a field I had strayed. Statistics of literary vocabulary proved to have their own special problems, their own peculiar difficulties and sources of fallacy, which no one apparently had made any attempt systematically to explore.

We conclude this section with a quite different contribution of Quantitative Linguistics to Statistics. Back in the 10 th century, ibn Wahab al-Kātib had the following interesting remark on Arab phonology (Mrayati et al 2007, 110):

The closer the outlets of two letters, the heavier on the tongue in articulation. The Arabs tend by nature to use what is inherently light and easy to pronounce, and to avoid using all that is awkward. It follows that they hardly take up two letters of the same outlet or of two equal outlets. If, however, such letters happen to occur together, they tend to assimilate the one into the other.

That remark was not to be quantified until ten centuries later, with Zipf's work on the "law of least effort" [START_REF] Zipf | Human behavior and the principle of least effort[END_REF]. Nowadays, statistical applications of Zipf laws and the like (Benford law, Pareto law) are ubiquitous, from econometrics to physics, biology, and computer science [START_REF] Reed | The Pareto, Zipf and other power laws[END_REF].

Statistics will reach an accuracy level…

It cannot be asserted that letter counting has always been a central preoccupation to statisticians. Yet it was present at several crucial stages in the development of the field: the Arab beginnings, the organization into a new scientific discipline in the middle of the 19 th century, the early development of stochastic processes by Markov, and the creation of information theory by the cryptanalysts of WWII. These four periods will be examined successively.

As we have seen before, Arab cryptologists have made quite substantial letter countings (3667 letters for Al Kindī). They certainly deserve to be regarded as pioneers of Statistics, at least in the sense of data collection; presenting them as specialists of data treatment would be an overstatement. However, it must be noted that they did have a clear idea of variability and sample size. This will be illustrated by two quotations from 'Afīf ibn 'Adlān's treatise. The first one shows how he expressed letter frequencies, comparing them to the most frequent. Notice the last sentence of the passage (Mrayati et al. 2003a, 48).

As far as frequency of occurrence is concerned, some Arabic letters are abundant, and these are seven in number, i. [ā] occurred in a piece of writing as many times as, say, 600 times, it has been found as a corollary, that the letter [l] would occur about 400 times, add or take few, the letter [m] 320 times, the letter [h] 270 times, [w] 260 times, [y] 250 times, and the letter [n] 220 times. This is so in all likelihood, but there may be some variation in this order.

The second quotation from the same treatise gives a quantified indication on the length of the text (Mrayati et al. 2003a, 52).

The length of the text to be cryptanalysed should be at least in the neighborhood of ninety letters as a rule of thumb, because the letters thus would have had three rotations. Yet, the number of letters may be less than that in certain cases.

Arguably, the organization of Statistics into a new branch of Science can be dated to a visit of Adolphe Quetelet to London in 1933, and his meeting with several statisticians there [START_REF] Hacking | The Taming of Chance[END_REF], 61, Babbage 1864, 434, Quetelet 1873, 156). Among them, Charles Babbage (1781Babbage ( -1871)). Quetelet and him had made friends at a dinner in Paris, in 1826 (Quetelet 1873, 152), and they had regularly corresponded since. An interesting character with multiple interests, Babbage is best known nowadays for his "Analytical Engine", the ancestor of computers. He also deserves a place in the histories of Cryptology and Statistics. In Cryptology, he is credited for being the first to have cracked the famous Vigenère code (Mollin 1990, 74). His own feelings were ambivalent: "Deciphering is, in my opinion, one of the most fascinating of arts, and I fear I have wasted upon it more time than it deserves" (Babbage 1864, 235). During that "wasted time", he acquired a solid experience of lexicography, counting and sorting letters and words. In Statistics, Babbage was a passionate supporter of the "Law of Averages." He proposed to gather in a huge set of data the "Constants of Nature and Arts", that he had grouped in 19 chapters. Among these constants, a good number were of statistical nature (Hacking 1990, 52). Chapter 18 of the list certainly owes much to his experience as a cryptanalyst (Babbage 1832, 338).

18. Tables of the frequency of occurrence of the various letters of the alphabet in different languages, -of the frequency of occurrences of the various letters at the beginnings or endings of words, -as the second or as the penultimate letters of words, -of the number of double letters occurring in different languages, -of the proportion of letters commencing surnames amongst different nations.

It was more than a vague intention: on the same year Babbage sent to Quetelet a frequency table of double letters in 10,000 words of 7 different languages (Quetelet 1832, 135-136). Twenty years later, during the preparation of the first international congress, he insists again (Quetelet 1873, 162):

On reading the printed views of your statistical congress, it appears to me that the constants of nature are within your limits, and I am more than ever convinced that even an attempt to collect a portion of them would contribute largely to the advancement of knowledge and to the economy of the time of its cultivators.

Babbage devoted one of his talks of 1853 in Brussels to the Constants of Nature and Arts, and Quetelet agreed to reproduce [START_REF] Babbage | On the Advantage of a Collection of Numbers, to be Entitled the Constants of Nature and Art[END_REF] in the proceedings (Quetelet 1853, 226). Quetelet fully supported Babbage's views on the law of averages: his own theory of the "Average Man" had been conceived before 1831 (Stigler 1999, 51-65). As already said, Quetelet had himself counted letters in 1828. Yet, he never was really convinced. Forty years later, he remembered the episode more like one of Babbage's quirks, than a true contribution to the Advancement of Knowledge.

We have already mentioned, at the beginning of this article, the singular tendency that our friend had, to deal more specially with questions that others lost sight of. I thought appropriate not to conceal from him a remark that I had already made in this respect; presumably, my observations made him smile, and he transmitted to me a new writing on the subject. That time, he had lost sight of high speculations of political economy: he had dealt with researches that he had made on the number of times that letters are doubled in a word, for instance on ten thousand different words. It may be curious to see the results that he got, comparing the English, Italian, German, and Latin languages. But what advantage can be expected in general from such a study? Admittedly, direct profit for the happiness of mankind was remote; but one cannot help wondering whether by counting more letters, Quetelet and Babbage would have realized earlier, how variable some "Constants of Nature and Arts" can be.

Neither of them seems to have been aware of the work of Viktor Yakovlevitch Bunyakovsky (1804-1889). The contributions of this Russian mathematician to Statistics deserve a more important place than they have been given so far, in particular in connection to our subject. In 1847, Bunyakovsky published in a literary journal an article entitled "On the possibility of introducing definite measures of confidence in the results of some observation sciences, mainly statistics" (Petruszewycz 1979b, 47). As the title says, confidence intervals are rigorously defined, and Bunyakovsky concludes with a prophetic vision of the discipline.

One can nearly predict that, some time from now and perhaps very soon, Statistics, following observation sciences, will reach an accuracy level that, while remaining by essence below that of astronomic results, will still be way above what it is in the present state of that science.

As Bunyakovsky himself says, he could have terminated his article there. But he goes on to present a new application.

The new application relates to grammatical and etymological research, as well as comparative philology. At first glance, such investigations seem alien to mathematical analysis, nevertheless it can be said with certainty that a huge field for rigorous mathematical applications lies there. My assertion is not based upon more or less rigorous assumptions and conjectures, but upon a critical examination of the discipline, upon some trials that I have already realized, and upon analytical formulas that I have deduced to define numerical probabilities of various linguistic derivations.

Bunyakovsky then goes on, giving examples of assertions that could be submitted to rigorous statistical proof, detailing data that should be collected on languages, and finally, advocating a collaboration between mathematicians and philologists. He also mentions the possibility to publish his own research on the subject, "should another opportunity arise"; unfortunately, no such work of his has been found to this date. Nevertheless, he not only paved the way for rigorous assertions in statistics, grounded on precise confidence intervals, but he also set up an even more ambitious program than that of Lieber: Bunyakovsky deserves to be counted as well among the founders of Quantitative Linguistics.

When Andrei Andreievich Markov (1856Markov ( -1922) ) began working on his chains of events in 1906, his motivation was mathematical: he wanted to prove that stochastic independence is not a necessary condition for the law of large numbers. Yet the only illustration that he ever gave was again the proportion of vowels to consonants: he counted 20,000 letters in Pushkin's "Eugen Onegin", 100,000 letters in Aksakov's "Childhood years of Bagrov's grandson", and made a very thorough statistical study of his counts, proving in particular that vowels and consonants did not alternate independently. The mathematical motivations of Markov, the connection of his work with cryptology and linguistic studies of his time, have all been amply documented, by Micheline Petruszewycz (Petruszewycz 1979a[START_REF] Petruszewycz | Chaînes de Markov et statistiques linguistiques[END_REF], and more recently by David Link (Link 2006a(Link , 2006b) ) who also provided an English translation of Markov's famous article on Eugen Onegin, together with Gloria Custance [START_REF] Markov | An example of statistical investigation of the text Eugene Onegin concerning the connection of samples in chains." Translated by Gloria Custance and David Link[END_REF]. The reason why Markov chose his linguistic application has also been thoroughly discussed: Micheline Petruszewycz convincingly argues that Bunyakovski's article may have been a source of inspiration (Petruszewycz 1979b). The obvious vowel-consonant filiation from Alberti to Markov through Pughe and Lieber, should not hide the major shift inaugurated by Markov in the mathematical modeling of language. From Markov on, a succession of letters would be seen as a dynamic source of information, i.e. as a stochastic process. The approach is perfectly explained in [START_REF] Shannon | A mathematical theory of communication[END_REF], the founding article of information theory by Claude E. Shannon (1916Shannon ( -2001)). There, Shannon proposes several models of increasing likelihood and information content, for the random generation of texts. He explicitly refers to Markov chains, andcites Maurice Fréchet (1878-1973), who helped popularize Markov's ideas in the west (Link 2006b). Shannon later detailed the application of his information theory to Cryptology [START_REF] Shannon | Communication theory of secrecy systems[END_REF] and Linguistics [START_REF] Shannon | Prediction and entropy of printed English[END_REF]). The history of information theory has been accounted for in several works: (e.g. [START_REF] Pierce | The early days of information theory[END_REF][START_REF] Brillouin | Science and information theory[END_REF][START_REF] Cover | Elements of information theory[END_REF]. It would be simplistic to reduce this major landmark of Statistics to the single influence of Cryptology. However, the deep involvement of some of the major actors into the war effort during WWII cannot be downplayed [START_REF] Kahn | The codebreakers -The Story of Secret Writing[END_REF][START_REF] Mollin | Codes: the guide to secrecy from ancient to modern times[END_REF]. Not only Shannon himself, but also Solomon Kullback (1907Kullback ( -1994)), and Alan M. Turing (1912Turing ( -1954) ) were deeply involved as cryptanalysts throughout WWII. Turing is best known for his role in theoretical computer science; his statistical thinking has been analysed in [START_REF] Good | Studies in the history of probability and statistics. XXXVII A. M. Turing's statistical work in World War II[END_REF]). Kullback's experience as a cryptanalyst lead him to write one of the first modern textbooks of statistical Cryptology [START_REF] Kullback | Statistical methods in cryptanalysis[END_REF].

Conclusion: polymaths of past centuries

The excessive specialization of modern science has induced an anachronistic tendency to associate each name to a discipline: Al Kindī to Philosophy, Alberti to Architecture, Viète to Algebra, Lieber to Linguistics, Babbage and Turing to Computer Science, Markov to Probability, just to mention a few names from the previous sections. The contents of this article would probably have puzzled most of its actors, who perceived knowledge as a whole and viewed themselves as scholars, or philosophers, in a very broad sense of the word: see (Burke 2012, chapter 6) on the progressive division of knowledge during the second half of the 19 th century. The impressive variety of subjects treated by Al Kindī in his different treatises makes it meaningless to lock him into a single discipline (Mrayati et al. 2002, 78). The same can be said of Alberti [START_REF] Grafton | Leon Battista Alberti, master builder of the Italian Renaissance[END_REF], or Babbage [START_REF] Babbage | Passages from the life of a philosopher[END_REF].

Most of the characters of our story had a training in classics, and knew Latin and Greek. Significantly, when Quetelet reviews Hayez's thesis in (Quetelet 1829, 391-394), he relays the new doctor's complaint, about having to write in Latin on a subject unheard of in Rome. Poetry, particularly in Latin was familiar to all. A well known anecdote reports that Leonhard Euler (1707-1783), was able to recite Virgil's Aeneid from any page he was given the number of. The statistician Francis Y. Edgeworth (1845Edgeworth ( -1926) ) made a statistical study of the dactyls in the same Aeneid, at the time when authorship attribution studies were just beginning (Stigler 1999, 111).

Cryptography, if only as a hobby, has also been familiar to most educated people for a very long time. Many well known names could be cited, like Bacon, Newton, Casanova, Euler, as being more or less marginally associated to cryptograms. "The Doctor" John Wallis (1616-1703), epitomizes our subject. He was an active cryptanalyst during the Glorious Revolution: he left examples of deciphered correspondence, together with instructions, translated into English in [START_REF] Davys | An essay on the art of decyphering[END_REF]. He was one of the very best mathematicians of his time; in Statistics he contributed the first English book on probabilities, shortly before the introduction of the first mortality tables by John Graunt and Edmund Halley. Last but not least, he wrote the first grammar of the English language… in Latin.

Eclectism, love of humanities and taste for riddles, we believe that these were important instigators for the mutual enrichment of Cryptology, Quantitative Analysis, and Statistics, through letter counting. There may have been, at least in some cases, another marginal factor. Most of the figures cited in the article were professors as well. Cryptograms make excellent classroom exercises. Letter counting also is a good collaborative activity to make students understand data collection [START_REF] Richardson | Morse Code, Scrabble, and the Alphabet[END_REF]). We find it significant that Markov's second study on Aksakov only appeared in his textbook on probability (Link 2006a). Think of a time when some had to teach statistics without computers nor internet to download data sets from…

  e.: [ā] , [l], [m], [h], [w], [y], and [n]. If the letter