Carlos Yoong
email: carlos.yoong@mail.mcgill.ca

Anders Thorin

Mathias Legrand

The Wave Finite Element Method applied to a one-dimensional linear elastodynamic problem with unilateral constraints

Keywords:

The Wave Finite Element Method (WFEM) is implemented to accurately capture travelling waves propagating at a finite speed within a bouncing rod system and induced by unilateral contact collisions with a rigid foundation; friction is not accounted for. As opposed to the traditional Finite Element Method (FEM) within a time-stepping framework, potential discontinuous deformation, stress and velocity wave fronts are accurately described, which is critical for the problem of interest. A one-dimensional benchmark with an analytical solution is investigated. The WFEM is compared to two time-stepping solution methods formulated on a FEM semi-discretization in space: (1) an explicit technique involving Lagrange multipliers and (2) a non-smooth approach implemented in the Siconos package. Attention is paid to the Gibb's phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system. It is numerically found that the WFEM outperforms the FEM and Siconos solution methods because it does not induce any spurious oscillations or dispersion and diffusion of the shock wave. Furthermore, energy is not dissipated over time. More importantly, the WFEM does not require any impact law to close the system of governing equations.

Introduction

Collisions between structural components can be observed in a variety of industrial processes, such as rock breaking equipment, vehicles crash testing, drilling tools, and many others. Through contact forces preventing non-physical inter-penetration of matter, a collision initiates a disturbance in the form of a stress wave which travels away from the region of contact. Corresponding governing equations are derived by the laws of mass conservation, momentum conservation, and energy conservation. Additional complementarity conditions are incorporated to properly reflect the impenetrability of bodies. Analytical solutions are only known for simple cases such as longitudinal collision of simple rods or transverse impacts on beams. For more general and challenging configurations, the solution should be numerically approximated.

As numerical procedures produce approximate solutions, they inherently exhibit limitations that might be unacceptable. For instance, the solution may feature non-physical spurious oscillations, commonly known as Gibb's phenomenon. The approximate wave propagation velocity may also be different from its physical counterpart due to period lengthening and amplitude decrease resulting in dispersion and dissipation errors [START_REF] Ham | A finite element method enriched for wave propagation problems[END_REF]. These potential numerical discrepancies should be avoided since they may lead to wrong design decisions.

A considerable amount of research has been devoted to the development of numerical techniques which attempt to overcome the mentioned limitation [2]. These methods traditionally assume a solution that is separated in time and space, involving three ingredients [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF]: (1) spatial discretization, (2) enforcement of the unilateral contact constraint, and (3) time integration.

In structural dynamics, the most popular technique for spatial discretization is the Finite Element Method (FEM). The displacement u.M; t / of an arbitrary point M at time t is approximated by a finite sum u.M; t / D P i i .M /u i .t/ where each term of the sum is separated in space and time: u i .t/ is the participation of the shape function i .M / [START_REF] Hughes | The Finite Element Method: Linear Static and Dynamic Finite Element Analysis[END_REF]. The initial local governing equation commonly given as a partial differential equation is thus transformed into a set of coupled ordinary differential equations (ODE) which are solely time-dependent. For wave propagation problems, this form yields numerical difficulties because information propagates at an infinite velocity through the considered domain [START_REF] Bathe | Finite Element Procedures[END_REF].

Unilateral contact conditions are mathematically expressed as a complementarity relation between the contact force and the clearance (gap) separating mechanical bodies [START_REF] Jean | The non-smooth contact dynamics method[END_REF] which can equivalently be seen as a set-valued function [START_REF] Eck | Unilateral Contact Problems: Variational Methods and Existence Theorems[END_REF]. This mathematical framework drastically complicates the development of adequate and robust solution methods. The most popular approaches enforcing unilateral contact conditions are the penalty method [START_REF] Armero | Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems[END_REF], the Lagrange multipliers method [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF], and non-smooth techniques [START_REF] Brogliato | Nonsmooth Mechanics: Models, Dynamics and Control[END_REF].

The semi-discretized problem in space in the form of ODEs with unilateral contact conditions turns out to be ill-posed [START_REF] Eck | Unilateral Contact Problems: Variational Methods and Existence Theorems[END_REF]. In order to recover uniqueness of the solution, an impact law is incorporated to close the system of equations. This approach is mainly used by the rigid-body dynamics community [START_REF] Moreau | An introduction to unilateral dynamics[END_REF]. Researchers usually consider the Newton impact law which relates the velocity after the impact v C with the velocity before impact v written as v C D ev , where e is a non-negative parameter called the coefficient of restitution [START_REF] Brogliato | Nonsmooth Mechanics: Models, Dynamics and Control[END_REF]. For a non-zero coefficient, non-physical oscillations might arise in the displacement of the contacting surface [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]. Choosing e D 0 leads to a zero-velocity phase of the contacting surface but results in a non-acceptable loss of kinetic energy [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF].

Time integration of the semi-discretized problem can be performed in various ways which vary according to the approach considered for enforcing the constraints. The idea of numerical time integration is to compute the state of the system advancing from a set of initial conditions in a step-wise fashion through time, going to a future state at time t i C1 from a present state at time t i and separated by a time-step t [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. Classical time integration schemes in elastodynamics such as Newmark and HHT [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF] can be implicit or explicit procedures. In this framework, the formulations which make no explicit use of a restitution coefficient e essentially hide it within the procedure handling the contact constraints. It is known that the penalty approach possesses significant numerical limitations since the resulting ODE are known to be stiff and the time-step has to be very small [2,[START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. The integration in time of penalized models can instead be performed by solvers devoted to stiff ordinary differential equations [START_REF] Darvishi | The numerical simulation for stiff systems of ordinary differential equations[END_REF]. Newmark and HHT schemes can also be adapted to incorporate Lagrange multipliers [2]. The introduction of unilateral contact constraints in an implicit scheme may produce divergent solutions due to the additional high frequency content [START_REF] Krause | Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme[END_REF]. This can be overcome by adding high frequency numerical dissipation, as proposed in [START_REF] Deuflhard | A contact-stabilized Newmark method for dynamical contact problems[END_REF], however doing this causes conservative systems to dissipate energy [START_REF] Krause | Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme[END_REF]. Explicit schemes require small time-steps to ensure stability and an associated high computational cost for large-scale simulations [START_REF] Hughes | The Finite Element Method: Linear Static and Dynamic Finite Element Analysis[END_REF].

Time integration of non-smooth discretized formulation target systems where the contact constraint is treated by a nonsmooth approach, and are classified into: event driven methods and time-stepping techniques [START_REF] Paoli | Time discretization of vibro-impact[END_REF]. The event-driven methods are formally based on an accurate contact event detection mechanism and the variable time-step is adapted such that the end of the step coincides with such event. They accurately predict when a contact event occurs; nevertheless, they become inefficient when frequent transitions arise in a short period of time [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. Most of the implementations of these methods have dealt with discrete systems with a few degrees-of-freedom [START_REF] Studer | Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics[END_REF]. On the other hand, in time-stepping techniques, similar to the Newmark and HHT families mentioned above, detection of events is considered on the same step as the rest of the integration. They have been proven to be convergent and robust but a small time-step is often required [START_REF] Studer | Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics[END_REF]. Two main approaches have been proposed: (1) the Schatzman-Paoli scheme is based on a position formulation of the unilateral contact constraint [START_REF] Paoli | A numerical scheme for impact problems I: The one-dimensional case[END_REF] and (2) the Moreau-Jean scheme relies on a velocity formulation of the unilateral contact constraint renamed the velocity Signorini condition [START_REF] Jean | The non-smooth contact dynamics method[END_REF]. Recently, it was proposed to enforce contact conditions at the position and velocity levels simultaneously [START_REF] Üls | Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized ˛-scheme[END_REF]. This leads to minimal energy dissipation but non-physical numerical oscillations appear in the simulations.

In the present study, a technique called the Wave Finite Element Method (WFEM) [START_REF] Shorr | The Wave Finite Element Method[END_REF] is implemented in order to numerically solve a one-dimensional linear elastodynamics problem with unilateral constraints 1 . Within the framework of continuum mechanics, the WFEM discretizes the system into nodes and elements of finite length x. Then, contrary to traditional finite element techniques, the corresponding impulse-momentum principle is enforced iteratively at each node and element in order to establish a set of time-dependent algebraic relations that provide the mechanical state (i.e. displacements, velocities and forces) of all the elements and nodes. An elastic wave propagating at a finite speed c is reproduced through iterations in time advancing by a known prescribed time interval t D x=c.

A one-dimensional benchmark problem offering a known analytical solution is investigated. The WFEM is compared to (1) an explicit time-stepping technique combining the FEM with forward Lagrange multipliers [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF] and (2) a non-smooth approach implemented in the software Siconos [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. Attention is paid to the Gibb's phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system.

First, the benchmark problem is described. In a second section, the theoretical background of the WFEM is detailed but limited to the equations for the one-dimensional case only, and the appropriate enforcement of the contact conditions in the WFEM is described. Lastly, the results of the benchmark problem simulations are presented and discussed.

Benchmark Problem

In order to study the properties of the WFEM, a one-dimensional elastodynamics benchmark including unilateral constraints and exhibiting a closed-form solution is used in this work. It was first proposed in [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF] to explore the numerical properties of various time integration schemes.

The problem consists of a homogeneous elastic rod of length L and constant cross-sectional area A, as shown in Fig. 1, bouncing against a rigid foundation. Its initial displacement at time t 0 D 0 is u 0 .x/ and its velocity, v 0 .x/ where x 2 OE0 I L is the coordinate of a point of the rod. It is subjected to a constant external body force q 0 which can be seen as "horizontal" gravity here. The contacting end x D L of the rod is initially separated from the rigid foundation by a distance g 0 . Its mass density is and E stands for Young's modulus. An elastic wave will thus propagate at velocity c D p E= . The unknown displacement at position x in the rod and at time t is denoted by u.x; t/; quantity r.t / is the contact force. The gap function is defined as g.t / D g 0 u.L; t /. The formulation is then:

Local equation: R u Eu ;xx D q 0 ; 8x 2 0 I LOE; 8t > 0 (1)
where .R / is a second derivative in time while . / ;xx is a second derivative in space.

Boundary conditions at x D 0:

u ;x .0; t / D 0; 8t > 0 (2)
Complementarity conditions at x D L:

g.t / 0; r.t / 0; r.t/g.t/ D 0; 8t > 0 (3)
Initial conditions:

u.x; 0/ D u 0 .x/; P u.x; 0/ D v 0 .x/; 8x 2 0 I LOE (4)
This problem has a unique solution [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF] and the variation of the energy is equal to the work of the gravity force q 0 , 1 2

d dt ÂZ L 0 P u 2 C Eu 2 ;x dx à D Z L 0 q 0 P u dx; 8t > 0 (5)
The rod is initially at rest, that is u 0 .x/ D v 0 .x/ D 0, 8x 2 OE0 I L. By choosing a suitable set of parameters, the motion of the rod is periodic in time and known in a closed form.

The Wave Finite Element Method

This section introduces the background of the one-dimensional WFEM with emphasis on the appropriate enforcement of unilateral contact conditions. The solution algorithm is also provided. For a more detailed description of the method, the reader is referred to [START_REF] Shorr | The Wave Finite Element Method[END_REF][START_REF] Shorr | Analysis of wave propagation in elastic-plastic rods of a variable cross section using direct mathematical modelling[END_REF].

Formulation

The formulation targets longitudinal elastic wave propagation in a rod with the assumption of a known and constant wave velocity c. For t > 0 in the most general case, the rod is subjected to time dependent boundary forces F ḃ .t / together with the distributed load q 0 . The superscripts signs (C or ) refer to the location of the nodal force on the boundary, "negative" on the left side and "positive" on the right side, as depicted in Fig. 1. The rod

F b .t/ F C b .t/ L q.x; t/ D q 0 Figure 1.
Homogeneous continuous rod of interest is discretized into n elements of equal length x D L=n. For a homogeneous rod, the time t D x=c of wave propagation along each element is Quantities referring to the state of the elements will be labeled with a subscript j D 1; 2; : : : ; n. In addition, quantities referring to the two nodes associated to this element will be labeled with the same subscript j together with a superscript that defines the location of the node: "negative" sign on the left side of element j and "positive" sign on the right side. Lastly, external quantities acting on node j will be labeled with a subscript

F j F j 1 F j C1 Node j Node j 1 Node j C 1 Element j 1 Element j Figure 2.
External equivalent nodal forces j D 1; 2; : : : ; n C 1 and an "asterisk" (*) as superscript. The distributed load q 0 is replaced by equivalent nodal forces F j D q 0 x acting on nodes j D 1; : : : ; n C 1, see Fig. 2. On the boundary nodes j D 1 and j D n C 1, nodal forces induced by the distributed load take the following form:

F 1 D F nC1 D 1 2 q 0 x (6)
and the total nodal forces acting on the boundary of the discretized rod can be written as:

F 1 .t / D F b .t / C F 1 and F C n .t / D F C b .t / C F nC1 (7)
In Fig. 3, the discretized rod is depicted along with the boundary forces in Eq. [START_REF] Eck | Unilateral Contact Problems: Variational Methods and Existence Theorems[END_REF]. To perform numerical time integration, discrete The boundary nodal forces in Eq. ( 7) are approximated by step-wise values which are assumed to remain constant during a time-step t, and are denoted F 1 .t i / and F C n .t i /. Furthermore, it is assumed that every internal nodal quantity for j D 2; 3; : : : ; n (i.e. displacement, velocity and stress) remain constant during t, and are denoted v j .t i / and j .t i /. Only the states (i.e. velocity and stress) of the elements may vary during a time interval. Two adjacent elements j 1 and j are illustrated in Fig. 4. They are virtually separated for illustration purposes only. The law of momentum conservation at the connecting node j yields:

F 1 .t/ F C n .t / x
j j .t i / v j .t i / C j 1 .t i / v C j 1 .t i / Node j j 1 .t i / ; v j 1 .t i / j .t i / ; v j .t i / State of Element j 1 State of Element j
c.v C j 1 .t i / v j 1 .t i // D C j 1 .t i / j 1 .t i / c.v j .t i / v j .t i // D . j .t i / j .t i // (8)
where the pairs . j 1 .t i /; j .t i // and .v j 1 .t i /; v j .t i // are the stresses and velocities of the two elements at time t i . Moreover, 3 inner nodal equilibrium and continuity must be satisfied, that is:

j C j .t i / C j 1 .t i / D 0 Equilibrium v j .t i / D v C j 1 .t i / Continuity (9)
where j D F j =A. Substituting Eq. ( 8) into Eq. ( 9) leads to a relationship for the stresses and velocities at the internal nodes, for j D 2; 3; : : : ; n:

C j 1 .t i / D 1 2 j .t i / C j 1 .t i / C j C c.v j .t i / v j 1 .t i // v C j 1 .t i / D v j .t i / C C j 1 .t i / j .t i / =. c/ (10)
From Eq. ( 10), the state of each element at t iC1 is calculated as follows:

j 1 .t iC1 / D j 1 .t i / C C j 1 .t i / j 1 .t i / v j 1 .t iC1 / D v j 1 .t i / C v C j .t i / v j 1 .t i / (11)
The values computed by Eq. ( 11) are employed to determine the nodal quantities at time t iC1 denoted by C j 1 .t i C1 / and v C j 1 .t iC1 / using Eq. [START_REF] Brogliato | Nonsmooth Mechanics: Models, Dynamics and Control[END_REF]. Finally, Eq. ( 8) is applied to determine the boundary velocities by taking into account the states of elements j D 1 and j D n. For the given boundary stresses

1 .t i / D F 1 .t i /=A and C n .t i / D F C n .t i /=A, the velocities become: v 1 .t i / D v 1 .t i / 1 .t i / 1 .t i / =. c/; v C n .t i / D v n .t i / C C n .t i / n .t i / =. c/ (12)
Equations ( 9) to ( 12) establish a set of simple algebraic relations whose solution for each element and node, marching through time with prescribed time-step t, approximates a longitudinal elastic wave travelling in the rod.

Floating boundary conditions

Unilateral contact conditions are such that the contacting node shall either stick to the rigid foundation or remain free. The switch between the two configurations might take place during any time-step t. This contradicts the assumption of the WFEM which says that internal and boundary nodal quantities must remain constant during such time intervals in order to ensure energy and momentum conservation. This can be seen as the compensation of ensuring the exact wave velocity. To enforce contact constraints in the WFEM equations, Shorr developed a concept named Floating Boundary Conditions (FBC) [START_REF] Shorr | The Wave Finite Element Method[END_REF]. This concept relies on the prediction of the position of the contacting node. If the unilateral constraint of impenetrability is violated by the node during t, a temporary change is performed to the position of the rigid foundation into a local admissible position in order to avoid penetrating condition during timestep t. / > 0, "Situation 1" occurs and the computation procedure continues with a free node n C 1, as depicted in Fig. 6a. On the other hand, if g.t i C1 / < 0, a penetration is detected during the ongoing time-step: this is the "Situation 2" illustrated in Fig. 6b and a temporary change to the position of the rigid foundation needs to be performed into a local admissible position. To this end, a temporary gap g 0 .t i / is calculated if contact is assumed to arise at t i or a temporary gap g 0 .t iC1 / if contact is assumed to occur at t i C1 . Both cases are expressed as:

g 0 .t i / D g.t i / C g 0 .t i / at time t i g 0 .t i C1 / D g.t i / C g 0 .t i / at time t iC1 (15)
where g 0 .t i / denotes the distance that the rigid foundation has to undergo to be in a temporary admissible position. This distance is calculated by comparing jg.t i C1 /j with g.t i /:

1. If jg.t i C1 /j g.t i /, the rigid foundation virtually travels the distance jg.t iC1 /j such that g 0 .t i / D jg.t i C1 /j. Then, a virtual gap g 0 .t iC1 / is calculated with Eq. ( 15) and contact will occur at the subsequent time t iC1 , see Fig. 7a.

Solution Algorithm

The proposed solution algorithm 1 is a time marching procedure where Eq. ( 9) to [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] are solved iteratively at each node and element, starting from node and element j D 1. To simulate an elastic wave propagating with finite speed c, these equations are iterated in time advancing by a prescribed time-step t D x=c. Node n C 1 is the contacting node on which the FBC principle is implemented. The procedure at time t i can be divided in the following steps:

1. Calculation of the velocity v 1 .t i / through Eq. ( 12) and calculation of the velocity v 1 .t i C1 / and stress 1 .t i C1 / of element j D 1 for the subsequent time t iC1 using Eq. ( 11).

2. A recurrence procedure for j D 2; 3; : : : ; n is performed to calculate all nodal quantities and states of the elements, employing Eq. ( 9) to (11). end -end of floating boundary conditionsend Output: displacements, stresses, velocities, contact forces contact is activated, the position of the rigid foundation is temporarily changed and the velocity of the contacting node is set to v C n .t i / D 0. Otherwise, the calculation proceeds on node n C 1 with a free boundary condition.

Results

Dimensionless parameters identical to [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF] are considered and listed in Tab. 1. The rod is expected to have a periodic motion in time with periodic bounces against the foundation. The WFEM is compared to (1) an explicit time-marching technique combining the FEM with forward Lagrange multipliers [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF] and (2) a nonsmooth approach implemented in the package Siconos [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] The time-step of the FEM and Siconos satisfies the CFL condition [START_REF] Bathe | Finite Element Procedures[END_REF] t x=c. In Siconos, two coefficients of restitution are considered: e D 0 (inelastic) and e D 1 (perfectly elastic). Two computations were performed to evaluate the influence of the number of elements: (1) n D 100 and (2) n D 500 with corresponding time-steps listed in Tab. 2. Contact node displace- ments are shown in Figs. 8 and12, respectively. The spurious oscillations occurring at the first impact of the rod are depicted in Figs. 9 and 13. Energy levels normalized with respect to the exact constant energy are depicted in Fig. 14.

Coarse spatial discretization

The contact node displacement and energy behavior for n D 100 is depicted, respectively, in Fig. 8 and Fig. 14. FEM and Siconos (e D 0) present an unsatisfactory energy behavior due to dissipation when the contact node impacts the foundation; such behavior is aggravated with subsequent impacts. This is caused by the energy dissipation properties of the time integration schemes [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF][START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF]. In addition, FEM does not exhibit Gibb's phenomenon in the displacement approximation, as depicted in Fig. 9, which is due to the exact enforcement of contact constraints in displacement [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF]. Further, for Siconos (e D 0) spurious oscillations are not present in the displacement approximation since the node remains stuck to the foundation because of v C D 0; however, the complementarity impenetrability condition is violated. Siconos (e D 1) represents a perfectly elastic collision and the total energy is conserved, as shown in Fig. 14. Nevertheless, a coefficient e D 1 physically assumes that the velocity of the contacting node after the impact v C is the exact opposite of the velocity v before the impact: accordingly, the contacting node will hardly be in a constrained position on the foundation. This is reflected in the approximation of the contact node displacement which features non-physical spurious oscillations after the first impact as depicted in Fig. 9. The approximation of the contact node displacement deteriorates after several bounces of the rod. Contrary to the other methods, the WFEM does not dissipate energy during unilateral contact occurrences. Furthermore, the approximation of the contact node displacement does not present numerical oscillations during the collision with the foundation, and the unilateral constraint is exactly enforced as displayed in Fig. 9 where the contact node velocity goes from a non-zero velocity v to exactly a zero velocity during the contact phase, before it retrieves a non-zero post-contact velocity v C . In other words, it is numerically shown that the implemented formulation of the WFEM does not "hide" neither necessitate any impact law to be well-posed. The velocity and contact force are depicted in where the solution has Gibb's phenomenon in the free flying phase without any oscillations during the contact phase. Instead, the WFEM is capable of capturing discontinuities in the velocity field with no undesirable oscillations: this is true for all times.

Fine spatial discretization

By increasing the number of elements to n D 500, an improved displacement approximation is obtained for the FEM and Siconos (e D 0) solutions, see Fig. 12. Their energy behavior still presents dissipation as multiple impacts occur in time, as observed in Fig 14 . Siconos (e D 1) provides a displacement approximation similar to its counterpart but energy is not dissipated. In Fig. 13, Siconos (e D 1) still introduces non-physical oscillations on the displacement of the contact node: although not as pronounced as for n D 100, this badly affects the dynamics of the whole system. For e D 0, the contact position is erro- neously predicted which causes the contacting node to violate the complementarity conditions. For the WFEM, the time-step decreases as spatial discretization gets finer and again it is observed how the WFEM is capable of properly approximating displacements with higher accuracy than the other methods, and without energy dissipation as depicted in Fig. 14. This comes together with the fact that the unilateral constraint is exactly enforced without numerical oscillations.

Computation time

The computation time for each method is depicted in Fig. 15. All simulations were performed with parameters listed in Tab. 1 on a desktop computer with processor Intel Core i7-2600, CPU@3.4 GHz and 16 GiB of RAM memory. It is clear that the WFEM outperforms its competitors. From the results depicted in Fig. 14, it is evident that the approaches based on FEM discretization present numerical energy dissipation that clearly affects the approximate solution and provides an erroneous prediction of the rod dynamics. Increasing the number of elements reduces the numerical dissipation, at the cost of a higher computational effort.

Conclusions

This study focused on the application of the WFEM to a onedimensional elastodynamics problem subjected to unilateral constraints. The WFEM was compared to (1) an explicit timestepping technique combining the FEM with forward Lagrange multipliers and (2) a non-smooth time-stepping approach implemented in the software "Siconos". Attention was paid to the Gibb's phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system. Additionally, the computation time of each method is included.

From the results, it is clear that the WFEM properly captures unilateral contact induced waves propagating at a finite speed with discontinuous deformation, stress and velocity wave fronts. This approach provides an accurate approximation of the solution without presenting any spurious oscillations. Energy is not dissipated over time as opposed to what is observed with the FEM and Siconos solvers. These advantageous numerical properties are not affected as time increases and get improved with a finer spatial discretization. This numerically shows that FEM-based numerical approximations assuming a solution in the form of a finite sum u.M; t / D P i i .M /u i .t/ separated in space and time are inappropriate to accurately capture travelling waves, which is crucial in elastodynamics involving unilateral contact constraints. Additionally, the WFEM does not require any impact law to retrieve the exact solution as opposed to all FEM formulations. This is a spectacular outcome of the present study since these impact laws are questionable in the context of continuum mechanics.

These promising results should be complemented by further WFEM investigations in a multi-dimensional framework. This is possible by enforcing the impulse-momentum principle at the connecting nodes of adjacent elements, assuming small strain, and small element rotation. A discussion on two-dimensional problems can be found in [START_REF] Shorr | The Wave Finite Element Method[END_REF], however it is limited to rectangular domains and elastic materials. Further studies should also focus on the implementation of the WFEM in complex geometries, three-dimensional domains, and large deformations problems.

Figure 3 .

 3 Figure 3. Boundary forces acting on the rod

Figure 4 .

 4 Figure 4. Interaction between elements j 1 j time t i

Figure 8 .

 8 Figure 8. Contact node displacement for n D 100: Exact Sol. [ ], WFEM [ ], FEM [ ], Siconos (e D 0) [ ] and Siconos (e D 1) [ ]

2 ]Figure 9 .

 29 Figure 9. Gibb's phenomenon for n D 100: WFEM [ ], FEM [ ], Siconos (e D 0) [ ] and Siconos (e D 1) [ ]

Figure 10 .

 10 Figure 10. Contact node velocity for n D 100: Exact Sol. [ ], WFEM [ ], FEM [ ], Siconos (e D 0) [ ] and Siconos (e D 1) [ ] Figs. 10 and 11. Siconos (e D 1) presents unphysical oscillations during the contact phase and this behavior is aggravated with time. FEM and Siconos (e D 0) predict a similar response,where the solution has Gibb's phenomenon in the free flying phase without any oscillations during the contact phase. Instead, the WFEM is capable of capturing discontinuities in the velocity field with no undesirable oscillations: this is true for all times.

3 ]Figure 11 .Figure 12 .

 31112 Figure 11. Contact force for n D 100: Exact sol. [ ], WFEM [ ], FEM [ ], Siconos (e D 0) [ ] and Siconos (e D 1) [ ]

3 ]Figure 13 .

 313 Figure 13. Gibb's phenomenon for n D 500: WFEM [ ], FEM [ ], Siconos (e D 0) [ ] and Siconos (e D 1) [ ]

Figure 14 .Figure 15 .

 1415 Figure 14. Normalized energy for n D 100: FEM [ ], Siconos (e D 0) [ ]; n D 500: FEM [ ], Siconos (e D 0) [ ]; Siconos (e D 1) [ ]; WFEM [ ]

  Figure 5. approaching the rigid foundation at t i Consider element n of the rod impacting a rigid foundation as displayed in Fig.5. The gap between the rod and the foundation at instant t i is g.t i / and the gap at the subsequent instant t iC1 is g.t iC1 /. During the corresponding time-step t, four situations may arise: 1. Open gap remains, g.t i / > 0 and g.t i C1 / > 0 2. Open gap g.t i / > 0 changes to penetration g.t iC1 / < 0 3. Permanent contact g.t i / D g.t i C1 / D 0 4. Initial contact g.t i / D 0 changes to open gap g.t iC1 / > 0 An open gap occurs at t i , such that g.t i / > 0, as illustrated in Fig. 5. Accordingly, contact node n C 1 is free, that is C

		g.t i /
		g.t i / g.t i C1 /
		(a) Open gap
		g.t i /	g.t iC1 /< 0
		g.t i /
		(b) Penetration
		Figure 6. Possible contact configurations during time step t
		Situations 1 or 2
	Element n	g.t i /
		Node
		n C 1

n .t i / D nC1 . Employing Eq. (12) to compute the contact node velocity v C n .t i / yields: v C n .t i / D v n .t i / C . nC1 n .t i //=. c/ (13) The gap g.t i C1 / is then calculated through: g.t iC1 / D g.t i / C g.t i / with g.t i / D v C n .t i /t (14) where the two states g.t iC1 / > 0 or g.t i C1 / < 0 might arise. If g.t iC1

  2. If jg.t i C1 /j > i /, the rigid foundation virtually travels the distance g.t i /, hence g 0 .t i / D g.t i /. Consequently, contact arises at time t i and the virtual gap g 0 .t i / is calculated with Eq. (15), see Fig. 7b. In this case, calculation at time t i must be performed considering that the contacting node is not moving due to the contact condition, such that v C n .t i / D 0. The position of the rigid foundation can be subsequently restored when an open gap arises again; in such a case, g 0 .t i / D 0. g 0 .t i / D jg.t iC1 /j Temporary change to the position of the rigid foundation Situations 3 or 4 A contact occurs at t i such that g.t i / D 0.

	Change
	(a) Contact occurring at t iC1
	Then, assuming that contact remains during the time-step, the contact stress C n .t i / is calculated with v C n .t i / D 0 employing Eq. (12):
	C n .t i / D n .t i / where both C n .t i / < 0 (compression) and C cv n .t i / C nC1 n .t i / > 0 (tension) (16) can occur. If C

g 0 .t i / D g.t i / Change (b) Contact occurring at t i Figure 7. n .t i / < 0, the contact condition is true and the calculation can continue with v C n .t i / D 0. On the other hand, if C n .t i / > 0 a recalculation of the step considering a free boundary condition is performed.

  Simulation Procedure Input: number of elements n, total number of steps N , initial gap g 0 , boundary conditions at node 1 and n C 1, initial conditions, external load q 0 , wave velocity c for i D 0 W N [Time Loop] do Discrete time instant, t i D it ; Stress and velocity at node 1;

	Stress and velocity at element 1;
	for j D 2 W n [Element Loop] do Stress and velocity at node j 1;
	Stress and velocity at element j 1;
	Apply continuity and equilibrium;
	end
	-floating boundary conditions -
	if g.t i / D 0 then Contact occurs at node n C 1; Calculation of contact stress: C n .t i /; if C n .t i / > 0 then Free boundary condition at node n C 1; Calculate gap: g.t iC1 /; Change rigid foundation position;
	else
	end	Contact occurs at node n C 1;
	else
	Free boundary condition at node n C 1; Calculate gap: g.t i C1 /; Change rigid foundation position;

3. Assumed contact g.t i / D 0. The contact stress is calculated to determine if such an assumption is satisfied. If Algorithm 1:

Table 1 .

 1 2 . Dimensionless simulation parameters

	Parameter	Value
	Simulation Time	20
	Young's Modulus, E	900
	Density,	1
	Rod Length, L	10
	Initial Gap, g 0 External Body Force, q 0	5 10
	Wave Velocity, c D	p	E=	30

Table 2 .

 2 Time-step for simulations

	Method t [n D 100] FEM 10 3 Siconos 10 3 WFEM x=c = 3 10 3	t [n D 500] 10 4 10 4 x=c = 6 10 4

It should be noted that a different method under the same name Wave Finite Element Method was proposed by Mace and collaborators[START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF]. The WFEM considered in the present work has a completely different ground.

Moreau's Time Stepping Scheme based on the Â-Method is employed. For the proposed simulations, an implicit version is considered with  D 1=2.


Laurent Buisson

Jérôme Euzenat

A quantitative analysis of reasoning for RMSes

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Jérôme Euzenat

Contexts for nonmonotonic RMSes

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Laurent Buisson

Jérôme Euzenat

The ELSA avalanche path analysis system: an experiment with reason maintenance and object-based representations

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Jérôme Euzenat

Michel Le

Éric Mazeran

Michel Weinberg

Generic embedding of an uncertain calculus in objects and rules

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Adam Smith's use of multiple references for his pin making example

knowing or asking what they contained. […] When the sixteen volumes of manuscript were burnt Smith's mind seemed to be greatly relieved." What did these 16 volumes contain? Why would Adam Smith want to keep part of his work secret ? Adam Smith used the pin maker example to advance his theory that the division of labour results in productivity improvements. The organisation of work within the small pin making industry was considered sufficiently convincing that it was used to defend an economic law. The division of labour is at the beginning of The Wealth of Nations.

While the study of pin making has been examined and commented at length, it still remains unclear as to where Adam Smith acquired his understanding of the pin industry.

One hypothesis is that Smith himself collected the data in the field. However, we can say neither when nor where he would have done so. A second hypothesis is that his data comes from his readings. It is necessary then to identify the one original text or several texts.

We suggest that Adam Smith based his work on several French texts. In the second half of the 18 th century, the Diderot's Encyclopédie and the Académie des Sciences de Paris explained pin manufacturing in detail. These texts were later copied, commented and synthesised. A large number of readings were thereby available to Adam Smith. Our thesis appears reasonable. It still remains to be shown however how these texts were used by Smith to construct his work.

Theories, past and present, explaining the origins of Smith's pin making example are first presented. They are then examined in light of the texts themselves. We notably highlight a number of weaknesses in the most often cited thesis, that of the use of the article « Épingles » from the Encyclopédie. We complete this thesis with three other documentary sources. We then show how Smith's writings are developed from theses four sources.

1) Ferguson's French source

Adam Ferguson was a colleague and friend of Adam Smith. They taught the same subject, moral philosophy, in Glasgow. In 1767 he published An essay on the history of civil society. The division of labour was considered in the fourth chapter : "Of the separation of Arts and professions". He wrote : "a people can make no great progress in cultivating the arts of life, until they have separated, and committed to different persons, the several tasks, which require a peculiar skill and attention". Progress comes not without division of labour. Such a division enables the simultaneous improvement of quality and productivity. "Every undertaker in manufacture finds, that the more he can subdivide the tasks of his workmen, and the more hands he can employ on separate articles, the more are his expenses diminished, and his profits increased". " By the separation of arts and professions, the sources of wealth are laid open" (p. 180 et 181).

Profit and wealth result from the division of labour.

An Adam Smith's thesis is exposed by his colleague before the publication of The Wealth of Nations in 1776. Adam Ferguson's approach was more a sociological one however. Excessive division of labour strains social ties. "The separation of professions, while it seems to promise improvement of skill, and is actually the cause why the productions of every art become more perfect as commerce advances; yet in its termination, and ultimate effects, serves, in some measure, to break the bands of society" (p. 218). This sociological thesis, opposed to the division of labour, would later be made famous by others, notably Karl Marx.

When Adam Smith discovered his colleague's writings, he immediately saw his own ideas. He accused Ferguson of copying. Ferguson denied any plagiarism. Both were inspired by the same source: "some unnamed French source "when Smith had been before him"" (Hamowy, 1968, p. 249 citing John Rae, 1895). Ferguson's defence leads us towards the French origins of Smith's ideas.

It may seem strange that Smith, who himself published in 1776, complained of plagiarism in 1767. Edwin Cannan (1896) provides an explanation. Notes taken in class by Smith's students were published. Cannan actually found the published text that had been sold to students in Glasgow. Smith spoke in class of the division of labour as early as 1763. This same subject could easily have been discussed with colleagues.

Adam Ferguson had the opportunity to copy Adam Smith. The question is whether he did or not. Was his reference to a common French source simply a way of hiding his own plagiarism? Cannan (1896) believes so and in so doing defends the originality of Smith's work. Others advance different hypotheses. Hamowy (1968, p. 255) argues that between 1780 and 1785 Adam Smith questioned draft texts from Adam Ferguson's book that would later be published after Smith's death in 1792. He notably gave the example of pin making.

In any case, the theory that French sources were used should not be discounted. What could this text have been? Many authors have debated this question.

August [START_REF] Oncken | Adam Smith und Adam Ferguson[END_REF] Germain Garnier adopted this point of view in his 1802 translation into French of The Wealth of Nations. This translation is still that used by French economics students.

The technical terms used by Delaire to describe the process are reintroduced in italics.

The first stage is "a man draws out the wire". Garnier boldly translated this phrase as « un ouvrier tire le fil à la bobille ». In French the word « bobille », derived from « bobine » (reel), is rarely employed. No dictionary suggests this same translation.

Garnier chose to do so as he read Delaire's text at the same time as he translated Smith's. He extrapolated. In his text, he added an additional step from the Encyclopédie « piquer les papiers » (to pierce the papers). He did more then than simply translate.

One man draws out the wire, another straights it, a third cuts it, a fourth points it, a fifth grinds it at the top for receiving, the head; to make the head requires two or three distinct operations; to put it on is a peculiar business, to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, … Un ouvrier tire le fil à la bobille, un autre le dresse, un troisième coupe la dressée, un quatrième empointe, un cinquième est employé à émoudre le bout qui doit recevoir la tête. Cette tête fait elle-même l'objet de deux ou trois opérations séparées : la frapper est une besogne particulière ; blanchir les épingles en est une autre ; c'est même un métier distinct que de piquer les papiers et d'y bouter les épingles ; enfin l'important travail de faire une épingle est divisé en dix-huit opérations distinctes ou environ … Hamowy (1968) also identified the Encyclopédie article as the French source of Smith's work. His main argument is that Adam Smith and Delaire both cite 18 pin making operations. The concordance of dates reinforces his thesis. We also know that Smith was in charge of purchases for the University of Glasgow library. He subscribed to Diderot's Encyclopédie. His colleagues criticised him as he used up their credits in doing so. Other works could not be purchased. Moreover, Smith reviewed the Encyclopédie quite favourably in the 1756 Edinburgh Review. Groenewegen (1977) offers a variant. "The famous pin example, probably derived from his observations of nail making in Kirkcaldy and his reading of the article, "Epingle", in Volume V of the French Encyclopedie." Adam Smith was born in Kirkcaldy. He lived there until the age of 14. He returned for two years at the end of his studies. It is quite possible that he observed the town's nail making activities. A possible rebuttal is that he would have looked up the word « clou » (nail) in the Encyclopédie instead and in so doing he would have found a different 7 column article. It is not so simple to pass from nails to pins.

Murray Rothbard (1995, p. 443) also defends this thesis. "There is strong evidence that the "French source" for both writers was the article Epingles in the Encyclopédie".

The famous article « Épingle » (Pin) was written by Alexandre Deleyre1 (1726-1797). A student of the Jesuits in Bordeaux, Deleyre worked as a poorly paid freelance journalist for diverse journals such as that of Diderot. He published several unremarkable works. His first on Francis Bacon was full of errors. The Critical Review reported "Our analyst has etablished absolute contradictions ... The analyst has confounded and obscured" (1757, 1 th semester, page 465). Deleyre ended his career as a member of both the Convention and the Institute2 . This is today's official and largely accepted version of Smith's sources. There are however several weaknesses to this theory:

-Adam Smith's first stage is that of wire drawing. "One man draws out the wire". The first stage in the Encyclopédie involves that "to yellow the brass wire"3 . Wire drawing, « tirer à la bobille », takes place at the second stage.

The texts differ on this first stage, yet they both refer to 18 stages in all.

-Adam Smith is unsure of the number of stages required to make the pin head. Is it two or three? "To make the head requires two or three distinct operations." The Encyclopédie clearly describes three.

-Adam Smith describes a strange fifth stage: "a fifth grinds it at the top for receiving the head." Why grind the pin head? The rounded end would be flattened. According to the Encyclopédie the points are finely reworked.

These activities are completely different.

-Adam Smith estimates production at 4800 pins per person, per day. The Encyclopédie doesn't estimate production rates.

The Encyclopédie appears a difficult reference to use as it covers so many subjects.

Why would Smith have thought of looking through the Encyclopédie for such a minor article? One rarely thumbs through a dictionary. A dictionary is typically used for a more deliberate inquiry about a precise topic. Why then was Smith so interested in pin making?

In light of this discussion, Germain Garnier's evidence appears relatively weak. The article « Epingle » from the Encyclopédie was no doubt influential but it is unlikely to have been the only source. Other texts may have come between Adam Smith and the Encyclopédie. This is the hypothesis we defend here.

3) The multiple references hypothesis While Adam Ferguson only spoke of one French reference that he shared with Adam Smith, the latter may have used several texts. He may only have shown one to his friend and gathered others as he worked on the subject.

The multiple references hypothesis is suggested by the following parts of his work:

-"The division of labour has been very often taken notice of, the trade of the pin-maker". Today's reader may be surprised by this remark. Were pins so often spoken at the time? What does "very often" mean?

-"Making a pin is divided into about eighteen distinct operations". Why is he uncertain as to the exact number of activities involved? The Encyclopédie clearly spoke of 18 operations.

The multiple references hypothesis is also suggested by the differences between the texts of The Wealth of Nations and that of the notes taken between 1762 and 1764 by his Glasgow students (Lectures on jurisprudence). He already spoke in class of the division of labour and pin making. We have a written trace of Adam Smith's thinking prior to

The Wealth of Nations. The differences between these texts suggest that he collected new data between the two publishing dates.

In the 1763 lectures, he estimated productivity at 2000 pins per day and per person.

In 1776, the level of production was 4800 pins per day and per person. This difference could be explained by the collecting of new and better information in the interim. He implies that the data were collected personally. "I have seen a small manufactory of this kind". He may also have uncovered a new reference. This second hypothesis is now developed.

What could these references have been? The French were also interested in pins at Pin making was the subject of these three important texts, published in 1755, 1761 and 1765. A great number of minor works were also published. Many periodicals included book reviews. L'art de l'épinglier was reviewed several times. The volumes of the Encyclopédie were mentioned and its articles were reviewed from time to time.

Journalists generally preferred philosophical articles. They were not really interested in pins.

Another minor text was the pin making article in technical dictionaries. The Encyclopédie had opened the dictionary market. As the Encyclopédie remained expensive, publishers filled the segment with cheaper, more specialised dictionaries.

These small "portable" dictionaries were actually compilations of more voluminous works otherwise inaccessible to the general public.

All the French texts are not cited here. Many could not have been used by Smith: the publishing dates do not match, the texts differ too widely, or they were not available in Glasgow. The French texts tended to copy one another. A detailed study shows that all the publications are based on two main handwritten studies in Normandy in 1717 and 1739. This subject will be examined in a forthcoming article.

In order to explore the multiple references hypothesis all the texts were collected and examined to determine how they could explain Adam Smith's writings. Four major texts were identified. They will now be presented individually.

3.1.) Dictionnaire portatif des arts et métiers by Macquer

Adam Smith owned a large collection of books (Bonar, 1894). He had bought many different works during his travels in France. One purchase is particularly important here.

The Portable dictionary of arts and crafts including etc4 , by Philippe Macquer, published in 1766 in Paris. This dictionary contained an article entitled « Epinglier » (Pin-making). Macquer is more precise and concise than the Encyclopédie. He gives work rates for eight workstations: 120 000 pins per day for the straightener (le dresseur), 30 000 for the putter (la bouteuse), and between 8 000 to 9 000 for the head maker (l'entêteur) (average of 8 500). The rhythms are different for each stage. We can complete those given by assuming that the grinder works at the same rate as the head maker. Adam Smith deduced an average production rate. Two calculations are possible: inverse the total of all inverted rhythms, or calculate the total of each execution rate. The second approach is the simplest. This calculation is straightforward. Two stages are omitted however: "to white pins in the boiler" (« blanchir les épingles dans la chaudière ») and putting them through the rub machine (« frottoire »). Macquer didn't give the rhythms for these stages. We hence have ten stages. This may explain why Adam Smith spoke of ten workers in The Wealth of Nations: "I have seen a small manufactory of this kind where ten men only where employed".

The other calculation, which involves inverting the rhythms, requires no assumption as to the length of the working day. In conclusion, Adam Smith is likely to have used Macquer's text as a reference for

The Wealth of Nations. He personally owned a copy. One can use this text to calculate the production rate of 4800 pins per day and per person. The rhythms differ par stage.

Smith averaged these differences out. Adam Smith may "have seen" this "small manufactory" "where ten men only were employed" within this text. Ancient Greek authors spoke of the division of labour as did English authors at the end of the 17 th century and the beginning of the 18 th . We now examine the terms they employed.

Firstly, in The Republic (Book II), Plato explains the gains through trade specialisation.

"Adeimantus -And will you have a work better done when the workman has many occupations, or when he has only one? Socrates -When he has only one.

Adeimantus -Further, there can be no doubt that a work is spoilt when not done at the right time?

Socrates -No doubt.

Adeimantus -For business is not disposed to wait until the doer of the business is at leisure; but the doer must follow up what he is doing, and make the business his first object.

Socrates -He must.

Adeimantus -And if so, we must infer that all things are produced more plentifully and easily and of a better quality when one man does one thing which is natural to him and does it at the right time, and leaves other things.

Socrates -Undoubtedly.

Adeimantus -Then more than four citizens will be required; for the husbandman will not make his own plough or mattock, or other implements of agriculture, if they are to be good for anything. Neither will the builder make his tools --and he too needs many; and in like manner the weaver and shoemaker.

Socrates -True.

Adeimantus -Then carpenters, and smiths, and many other artisans, will be sharers in our little State, which is already beginning to grow? Socrates -True.

Adeimantus -Yet even if we add neatherds, shepherds, and other herdsmen, in order that our husbandmen may have oxen to plough with, and builders as well as husbandmen may have draught cattle, and curriers and weavers fleeces and hides, --still our State will not be very large.

Socrates -That is true; yet neither will it be a very small State which contains all these.

Adeimantus -Then, again, there is the situation of the city --to find a place where nothing need be imported is well-nigh impossible.

Socrates -Impossible.

Adeimantus -Then there must be another class of citizens who will bring the required supply from another city? Socrates -There must.

Adeimantus -But if the trader goes empty-handed, having nothing which they require who would supply his need, he will come back empty-handed.

Socrates -That is certain.

Adeimantus -And therefore what they produce at home must be not only enough for themselves, but such both in quantity and quality as to accommodate those from whom their wants are supplied." (Plato, The Republic, Book II, translated by Benjamin Jowett) Xenophon goes even further. He speaks of the division of labour within the one trade, shoe making. It was an important trade for the city of Athens that imported skins and widely exported the shoes it produced. "In the great cities, owing to the wide demand for each particular thing, a single craft will suffice for a means of livelihood, and often enough even a single department of that; there are shoe-makers who will only make sandals for men and others only for women. Or one artisan will get his living merely by stitching shoes, another by cutting them out, a third by shaping the upper leathers, and a fourth will do nothing but fit the parts together. Necessarily the man who spends all his time and trouble on the smallest task will do that task the best" (Xenophon, Cyropaedia, book VIII, translated by Henry Graham Dakyns, edited by F.M. Stawell, Project Gutenberg Release).

Task specialisation occurred in major industries and in large towns. It depended on the market. On the other hand, "in a small city the same man must make beds and chairs and ploughs and tables, and often build houses as well; and indeed he will be only too glad if he can find enough employers in all trades to keep him. Now it is impossible that a single man working at a dozen crafts can do them all well" (ibidem).

Adam Smith probably knew of these texts. His writings are similar : "Country workmen are almost everywhere obliged to apply themselves to all the different branches of industry that have so much affinity to one another as to be employed about the same sort of materials. A country carpenter deals in every sort of work that is made of wood: a country smith in every sort of work that is made of iron. The former is not only a carpenter, but a joiner, a cabinet-maker, and even a carver in wood, as well as a wheel-wright, a plough-wright, a cart and waggon maker." (The Wealth of Nations, Book 1, chapter 3).

Through the ages task specialisation has been associated with high production volumes and quality. The term "division of labour", however, had never been employed.

Smith's English predecessors spoke of this same concept (Cf. [START_REF] Schumpeter | History of economic analysis[END_REF], pp. 56, 214, 373-376, Rashid, 1998). They identified and named it in various ways. Table 3 lists some of the terms employed. In the richer country, where the demands are great and constant, every manufacture that requires various processes, and is composed of different parts, is accordingly divided and subdivided into separate and distinct branches.

Table 3. The terms employed to describe the concept of the division of labour prior to Adam Smith's work.

Each author described the concept in his own way but all lacked a concise term. The verb "to divide" was often employed in two derived forms, "divided" and "dividing".

The substantive form was close. Adam Smith employed "division of labour", a term that refers both to the result (work is divided and everyone has their own speciality) and to the action itself (work is divided again and again, with more and more workstations created). In addition, Duhamel refers to the rapidity of work. Adam Smith develops this same idea as the "productive power of labour".

Smith may have been influenced by

We have now identified two sources, Macquer and Duhamel du Monceau. These texts do not explain however why Adam Smith was so interested in pin making. What event triggered his interest? A periodical may have set off his interest. The hypothesis here is that it was the Journal des sçavans.

3.3.) The Journal des sçavans (Scientists' Journal)

The Journal des sçavans was a monthly magazine, published from 1665 onwards in Paris. During the 18 th century it was closely associated with the Académie des Sciences.

It reported on the Académie's publications and published the minutes of the Académie's meetings. The University of Glasgow library held a subscription.

In its November 1761 edition, the Journal reviewed Duhamel's L'art de l'épinglier Tailoring: "cloth must be cheaper made, when one cards, another spins, another weaves, another draws, another dresses, another presses and packs; than when all the operations above-mentionned were clumsily performed by the same hand" (Petty, 1690, in Hull, 1899, T 2 p. 473),

(
Watchmaking: "if a man shall make the wheels, another the spring, another shall engrave the dialplate, and another make the cases, then the watch will be better and cheaper, than if the whole work be put upon by any one man" [START_REF] Petty | Another essay in political arithmetic[END_REF] After all, his family name does signify "metal worker".

4.2.) Interpretation and use of the Journal des sçavans

The multiple references hypothesis starts with these preliminary activities. Adam

Smith regularly read periodicals and the Journal des sçavans was one such publication that would have regularly published articles of interest. In 1761 he could have discovered the review of Duhamel's L'art de l'épinglier (Art of pin-making). The numerous steps in pin making would have interested him. The description was original.

Neither Petty nor Xenophon had used this example. Adam Smith also recognised the negative social consequences of the division of labour. "The man whose whole life is spent in performing a few simple operations ... has no occasion to exert his understanding ... He ... generally becomes as stupid and ignorant as it is possible for a human creature to become ... His dexterity at his own particular trade seems, in this manner, to be acquired at the expense of his intellectual, social, and martial virtues". (The Wealth of Nations, Book V, chapter I, part 3 article II).

On the division of labour's more technical aspects, Smith erroneously described the point sharpening on double headed pins. The French journal described this step rather ambiguously. The traduction 11 , could be understood as meaning that the pin was sharpened at both ends. Yet the text referred specifically to the « tronçons », or sections, of two pins. The labourer worked on longer pieces of metal that he could more easily hold between his fingers. He then cut the section into two separate pins.

Adam Smith ignored the difference between the sections and the pins. He wrote "a fifth grinds it at the top for receiving the head". He never mentioned the cutting of the section into two pins (the Encyclopédie's 7 th activity). Maquer's text was just as confusing and no doubt comforted Smith in his interpretation. Adam Smith presented these results in a strange way. He took the weight into account. "They could, when they exerted themselves, make among them about twelve pounds of pins in a day. There are in a pound upwards of four thousand pins of a middling size? Those ten persons, therefore, could make among them upwards of fortyeight thousand pins in a day. Each person, therefore, making a tenth pert of forty-eight thousand pins in a day." The production was weighed daily: 12 pounds of 4000 pins.

Smith's use of weight in his measure of production is difficult to understand. There were pins of different sizes. While the number of pins was not affected by size, the total weight of the pins was. Production time does not vary with pin size whereas it does with weight. According to Perronet there were 13000 number V pins per pound, 4000 number X pins, and only 1350 of the biggest size, number XXII. According to pin size, weight may vary tenfold, while work time varied little. Smith's idea of a pin "of a middling size" is imprecise. The most sold pins were the smaller sizes, number IV or V.

Why did Smith measure production rates based on weight when production time depended on the number of pins and not on their weight? One possible hypothesis is that he may have done so to hide his use of the French texts. This is difficult to verify. In any case he gave a production rate equivalent to that of the French texts, 4800 pins per day and per person.

4.6.) Summary of Adam Smith's references Adam Smith had access to four French texts, the Journal des sçavans, the Encyclopédie, Duhamel, and Macquer. These texts are coherent with the evolution of the different versions of his writings. We can match his ideas with those developed in these texts.

Tables 5 and6 Lectures on Jurisprudence, Report dated 1766 (p. 539) "This must be much more the case when a person's whole attention is bestowed on the 17 th part of a pin" 17 parts of the text L'art de l'épinglier de Duhamel, 1761 Lectures on Jurisprudence, Early draft (p. 566) "One man straightens the wire, another cut it, a third points it, a fourth grinds it at the top for receiving the head, there or four people are employed about making the head, to put it on is the business of a particular person, to gild the pins is the occupation of another, it is even a trade by itself to put them in the paper." « 3° On dresse le fil, 4° on coupe la dressée, 5° on empointe, 6° on repasse, 8°, 9°, 10°, 11° on frappe les têtes, 13° on blanchit les épingles, 18° on boute les épingles. » Encyclopédie, Delaire, 1755 "When this small operation is in this manner divided among about eighteen persons, these eighteen will perhaps among them make upwards of thirty six thousand pin a day." « une épingle éprouve dix-huit opérations» « il le faut bien pour piquer 36 000 épingles par jour » Encyclopédie, Delaire, 1755 "and those employed in every different branch of the work can often be collected into the same workhouse, and placed at once under the view of the spectator."

The 7 illustrations show workers labouring in the same room.

L'art de l'épinglier de Duhamel, 1761 "To take an example, therefore, from a very trifling manufacture; but one in which the division of labour has been very often taken notice of, the trade of the pin-maker;" Multiple references exist "One man draws out the wire" « la première opération consiste à passer le fil de laiton à la filière » Journal des sçavans, 1761 "another straights it, a third cuts it, a fourth points it, a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on, is a peculiar business, to whiten the pins is another; it is even a trade by itself to put them into the paper;" « 3° On dresse le fil, 4° on coupe la dressée, 5° on empointe, 6° on repasse, 8°, 9°, 10°, 11° on frappe les têtes, 13° on blanchit les épingles, 18° on boute les épingles. » Encyclopédie, Delaire, 1755 "and the important business of making a pin is, in this manner, divided into eighteen distinct operations" « une épingle éprouve dix-huit opérations» Encyclopédie, Delaire, 1755 "I have seen a small manufactory of this kind where ten men only were employed, and where some of them consequently performed two or three distinct operations." "They could, when they exerted themselves, make among them about twelve ponds of pins in a day. There are in a pound upwards of four thousand pins of a middling size. Those ten persons, therefore, could make among them upwards of forty-eight thousand pins in a day. Each person, therefore, making a tenth part of forty-eight thousand pins, might be considered as making four thousand eight hundred pins in a day."

Production rate calculated at 4 800 pins per day and per person.

Dictionnaire portatif des arts et métiers, Macquer, 1766

Table 6. Similarities between The Wealth of Nations and the French references

The multiple references hypothesis is a feasible one. The references were accessible to Smith, the dates match and the ideas are similar. Does this suffice? Any further proof was lost with the destruction of Smith's personal notes.

If we accept the hypothesis that Smith found inspiration in the four French texts, one question still remains unanswered. The texts are very rich. They provide a great deal of detail that was not exploited. For example, in Duhamel's text, Perronet identified only 14 steps in the production process. This figure was never cited by Smith. Other details are also omitted. Why? There are two possible answers. Firstly, these details are not required to illustrate the phenomenon under study. Secondly, the complicated vocabulary used made these details difficult to understand.

Adam Smith sought to demonstrate that the division of labour improved production rates. He used the French texts only to describe the extent of the division of labour, expressed as the number of steps (17 or 18 depending on the reference), and the daily production rate per person (2000 or 4800 depending on the reference). He narrowed his reading of these texts as only these details counted.

Another reason that may have led him to narrow his reading was the deliberately esoteric technical vocabulary employed. Diderot chose this writing style and Duhamel copied him. The Journal des sçavans and Macquer, on the other hand were more readable. It's quite possible that Smith only read parts of these texts and skimmed over the more complicated sections. This assertion however cannot be proven.

5) The implications of the multiple references hypothesis

The comparison of the French and the English texts tend to confirm our hypothesis.

It is highly likely that Adam Smith used the Journal des sçavans, the Encyclopédie, Duhamel and Macquer to write up his pin making example to illustrate the division of labour.

This hypothesis is consistent with the way Adam Smith worked: wide general knowledge, interest in variety of disciplines, open to opportunities provided by current affairs, in-depth use of documentation, talent for interpreting and reworking available information. He worked hard on the pin making example. It was no doubt of great importance to him. He used at the beginning of his book and in such a way that it could not be ignored.

One could criticise him, as we would an author today, for not citing his sources. Yet he behaved like many intellectuals of his time, often forgetting to cite his references.

  has no doubt that the French source was Montesquieu. « … kann nicht wohl ein Zweifel darüber obwalten, dass er unter der älteren gemeinsamen Quelle Montesquieu verstanden wissen wolle, niemand anders. ». Onken however, did not directly refer to the texts. He based his reasoning on Ferguson's aquaintance with the French author. Charles de Secondat de Montesquieu (1689-1755) was known throughout Europe for his 1748 work L'Esprit des lois (The Spirit of Laws) . As no description of pin making appears in this text, we abandon this theory. Contemporary experts favour the article « Épingles » (Pins) from Diderot's Encyclopédie as the French source of Smith's work. 2) The « Épingles » (Pins) article from the Encyclopédie Volume V of Diderot's Encyclopédie, published in 1755, included an 8 column article by Delaire entitled « Épingles » (Pins). The similarities with Adam Smith's text indicate that this is the French source of his work.

  this time. As early as 1723, Savary included an article « Espingles » (Pins) in his Dictionnaire du Commerce (Trade Dictionary). He mainly detailed the organisation of the pin maker's guild. The major texts on the pin making process were published in the Encyclopédie; Deleyre's 1755 article, « Epingles », and an article by Perronet, « Epinglier » (Pin-maker), in the 1765 volume of plates. The Académie des Sciences de Paris also wrote on the subject in 1761. Duhamel du Monceau published L'art de l'épinglier (The art of pin-making), an 84 page collection of writings to which Réaumur, Perronet and Chalouzières each contributed. The texts were interwoven, with each describing a different stage of the production process. The credit given to each author showed the emerging concern for intellectual property rights. Given each contributor's renown, Duhamel probably had no alternative but to identify the source. While Réamur was no longer alive at the time his manuscript had been widely read. Perronet on the other hand was very much alive. In charge of the engineering school, he was an influential figure in the administration of the Ponts et Chaussées (roads and bridges).

Philippe

  Macquer was the brother of Pierre-Joseph Macquer (1718-1784), member of the Académie des Sciences, famous chemist, and Lavoisier's teacher. The Macquer family was of Scottish roots. The Dictionnaire portatif was a two volume abridged edition. It was based on La description des Arts & Métiers (The description of Arts and crafts), a series edited by Duhamel in 1761. His article « Epinglier » is derived from part of this same series, L'art de l'épinglier (Art of pin-making). He mainly used Perronet's text. Macquer described in a succinct fashion the same activities as Deleyre. The first stage, as in Adam Smith's text, is to draw out the wire. "Traduction » 5

  have rounded up this result to obtain 4800 pins per day and per person.

  This reference was published in 1766. It explains the difference between The Wealth of Nations published in 1776 and Smith's 1763 lectures. It remains to be shown why Smith referred to pin making in his 1763 Glasgow classes. 3.2.) L'art de l'épinglier (Art of pin-making) by Duhamel Following on from Réamur, Duhamel du Monceau, a member of the Académie des Sciences de Paris, was in charge of publishing the scientific work about the Arts & Crafts. This work had been dragging on since the end of the previous century. The Académie was embarrassed by its non scientific nature. Diderot had access to all the Académie's documentation from the beginning of the century onwards. He notably consulted the illustrations. The success of Diderot's Encyclopédie changed things for the Académie. It regained interest in its publishing project. La Description des Arts & Métiers was first published by the Académie des Sciences in 1761. One of the first texts was l'art de l'épinglier. It was held by the University of Glasgow library. Adam Smith could have read this text. We can link this work to the expression "division of labour". The concept was known to other authors although different words were used to describe it. Smith was the first to employ the expression "division of labour".

  Duhamel du Monceau who employed an equivalent expression. He wrote in the introduction to the Art of pin-making (L'art de l'épinglier) : traduction 6 . The french expression « la division de ce travail » (the division of this labour) could be interpreted as meaning "the division of the labour of pin-making" . Within the context of the sentence, this expression could also refer to the work schedule. The french word "travail" (labour, work, job, occupation) is polysemous. It can be interpreted both ways. The Duhamel's expression « division de ce travail » may have inspired Smith's own expression, "the division of labour". Duhamel's text went on to list the 17 numbered parts. The first 16 are steps in the production process. Duhamel concludes (p.3) traduction 7 .

  pages[745][746][747][748]. The text was based on Duhamel's introduction. The author listed the steps in the production process. traduction 8 . This was the first stage listed by Adam Smith in 1776.Eighten steps in all are given in this text: to draw (« passer à la filière »), to clean (« décaper »), to straighten (« dresser »), to cut (« rogner »), to pint (« empointer »), to point again (« repasser »), to cut the sections (« couper les tronçons »), to turn the head (« tourner les têtes »), to cut the head (« couper les têtes »), to put on the head (« brocher la tête »), to punch the head (« l'assujettir »), to yellow (« jaunir ») and to whiten (« blanchir »), amount to 13 activities. The author continues with 5 more operations the iron pins (« les épingles de fer »), the black pins (« les épingles noircies »), the two heads pins (« à deux têtes », the hairpins (« en pincettes ») and finshes with putting pins on the papers by twenty five (« l'arrangement des épingles par quarteron sur le papier »). The 18 steps of the pin making process were thereby available elsewhere than in Deleyre's text.The Journal des sçavans widely cites Duhamel's work, and in particular the production rates. Traduction 9 . The text is a summary full of details, all of which would have interested Adam Smith. It is reasonable to assume that he took notice of this text. 4) Hypothesis on the way Adam Smith wrote up his texts Having examined how each text may have contributed to Adam Smith's writings, we can now piece together their sequence of use. To begin with, Smith knew of the concept of the division of labour through his readings of classical Greek authors and past English scholars. It is likely that he regularly read scientific journals as they were published. He would have discovered the description of the production process in the Journal des sçavans. The example would stand out given his previous readings. He may 8 « La première opération consiste à passer le fil de laiton à la filière pour le calibrer » 9 « Toutes ces opérations s'exécutent, à la vérité, avec une célérité merveilleuse » then have spoken about it with Adam Ferguson. Smith would have sought additional references on the matter. In Duhamel's text he would discover the term « division de ce travail ». In Diderot's Encyclopédie all 18 activities were clearly set out. He would have all the information he needed to prepare his Glasgow lectures. During a later trip to Paris he may have found and purchased Macquer's Dictionnaire. The pin making article would have been of great interest. He would now be ready to write The Wealth of Nations. 4.1.) Prior work on labour specialisation Adam Smith knew his classics well. He most certainly would have read Xenophon's work on the division of labour by Athenian shoe makers. He would also have read previous British scholars. He would have known that Petty, Mandeville, Hume and Tucker all discuss the division of labour and that each uses examples to help explain the concept. Petty gave three examples of the division of labour (Hull, 1899) :

  Hull, 1899, T 1 p. 260), Sea transportation: "Dutch are enabled to convey goods cheaply by sea because they specialise each ship for a specific function, […] they can afford a particular sort of vessel for each particular trade" (Petty, 1690, in Hull, 1899, T 1 p. 261). These examples were later used by other authors. Smith's previous readings no doubt motivated him to look for other examples of work division. His childhood in the nail making town of Kirkcaldy may have influenced his choice of a metallurgical activity.

Adam

  Smith would have spoken to Adam Ferguson about this text. His colleague would perfectly understand the economical implications. He would even draw a number of sociological conclusions. The text itself begins this same analysis. Traduction 10 . The local laywer Chalouzières, consulted by Duhamel, gave a sympathetic eye witness account of factory workers toil. It would be the catalyst for Adam Ferguson's sociological analysis.

«

  Ils le passent à la filière […] on dresse le fil […] on la coupe en tronçons […] l'empointeur leur fait une pointe à chaque bout […] le repasseur […] le coupeur de hanses […] le coupeur de têtes […] l'entêteur, on blanchit, la bouteuse » soit dix opérations Dictionnaire portatif des arts et métiers, Macquer, 1766

Table 1 .

 1 Extracts from book 1, chapter 1, Recherches sur la nature et les causes de la richesse des nations, Adam Smith, 1802, translated into French by Germain Garnier (italics as per the original text).

Table 2 .

 2 Execution rates are the inverse of the rhythms. Assuming a 12 hour day, we have 12*60*60 = 43 200 seconds. Using Macquer's figures, we arrive at an average execution rate of 9.06 seconds per pin. If we round this result downwards to 9 seconds, the production rate is 4800 pins per day and per person (see table2). Execution time for each pin making operation according to Macquer's (1766) rhythms.

	Operator	Rhythm	Execution rate
		(pins per day)
	Straightener (Dresseur)	120 000	0,36 seconds
	Point maker (Empointeur)	72 000	0,6 seconds
	Grinder (Repasseur)	72 000	0,6 seconds
	Cutter (Coupeur de hanses)	190 000	0,23 seconds
	Head cutter (Coupeur de têtes)	144 000	0,3 seconds
	Head maker (Entêteur)	8 500	5,08 seconds
	Paper piercer (Percer les papiers)	96 000	0,45 seconds
	Putter (Bouteuse)	30 000	1,44 seconds
	total	4 800 	9,06 seconds per pin
			rounded off to 9 seconds

  10 « On fait aussi quelques remarques générales sur le métier d'épinglier ... Ce métier est trèsmal-propre & contraire à la santé. La rouille du laiton, qui est verd-de-gris, agit sur les ouvriers plus ou moins selon la place qu'ils occupent dans la fabrique ... Les empointeurs ne sont pas robustes, meurent pulmoniques & de bonne heure » 11 « les empointeurs forment les pointes sur les meules d'acier ... Des repasseurs ou finisseurs les adoucissent sur une autre meule plus fine. Lorsque les fils des tronçons sont ainsi appointis par les deux bouts, il faut les couper à la longueur des épingles ». « Appointis par les deux bouts » global production rate. He would have worked with the figures and found a production rate of 4800 pins per day and per person.

  compare Smith's descriptions of pin making with the French references we presented previously.

	Lectures on Jurisprudence, Monday, March 28, 1763 (pp. 341-342)	Original text	Reference
	"The division of labour amongst different hands can alone account for	« cette énumération nous fournira autant	L'art de
	this"	d'articles qui feront la division de ce	l'épinglier de
		travail »	Duhamel, 1761
	"an instance frivolous indeed, but which will illustrate it; this is the pin-	Description of pin making	Journal des
	making"		sçavans, 1761
	"one cutts the wire"	« 4° On coupe la dressée »,	Encyclopédie,
			Delaire, 1755
	"another sharps the one end for receiving the head"	« les fils des tronçons sont ainsi appointis	Journal des
		par les deux bouts »	sçavans, 1761
	"another sharps the one end for receiving the head"	« Il faut former des pointes aux deux bouts	L'art de
		de ces tronçons de fil »	l'épinglier de
			Duhamel, 1761
	"3 or 4 are employed in making the head"	« 8°. On tourne les têtes. 9°. On coupe les	Encyclopédie,
		têtes. 10°. On amollit les têtes. »	Delaire, 1755
	"one puts it one, another forms the point, another gilds and another papers	« 11° on frappe les têtes, 5° on empointe	Encyclopédie,
	them."	13° on blanchit les épingles, 18° on boute	Delaire, 1755
		les épingles. »
	"So that in the making of a pin there are about 18 persons employed.	« une épingle éprouve dix-huit opérations»	Encyclopédie,
	These in a day will make about 36000 pins, and this comes to the same	« il le faut bien pour piquer 36 000	Delaire, 1755
	thing as if each one made about 2000."	épingles par jour »
	Lectures on Jurisprudence, Report dated 1766 (p. 490)
	"the pin maker therefore divides the labour among a great number of	« 4° On coupe la dressée, 5° on empointe,	Encyclopédie,
	different persons, the cutting, pointing, heading, and gilding are separate	11° on frappe les têtes »	Delaire, 1755
	professions."
	"Two or three are employed in making the head"	« 8°. On tourne les têtes. 9°. On coupe les	Encyclopédie,
		têtes. 10°. On amollit les têtes. »	Delaire, 1755
	"one or two in putting it on, and so on, to the putting them in the paper"	« 11° on frappe les têtes, 18° on boute les	Encyclopédie,
		épingles. »	Delaire, 1755
	"…being in all eighteen. By the division every one can with great ease	« une épingle éprouve dix-huit	Encyclopédie,
	make 2000 a day."	opérations »	Delaire, 1755
		« il le faut bien pour piquer 36 000
		épingles par jour »

Table 5 .

 5 Similarities between Smith's previous texts (prior to The Wealth of Nations) and the French referencesInquiry into the nature and causes of the Wealth of Nations, Livre I, chapitre I, 1776

	Original text	Reference

The right spelling of this name.

French Academy

« on jaunit le fil de laiton »

Dictionnaire portatif des arts et métiers, contenant en abrégé l'histoire, la description et la police des arts et métiers, des fabriques et des manufactures de France et des pays étrangers.

« Les épingliers achètent le fil en bottes ; ils le passent à la filière pour lui donner la grosseur que doit avoir l'épingle ».

« Il n'y a personne qui ne soit étonné du bas prix des épingles ; mais la surprise augmentera sans doute quand on saura combien de différentes opérations, la plûpart fort délicates, sont indispensablement nécessaires pour faire une bonne épingle. Nous allons parcourir en peu de mots ces opérations pour faire naître l'envie d'en connoître les détails ; cette énumération nous fournira autant d'articles qui feront la division de ce travail » (p. 1)

« Toutes ces opérations s'exécutent, à la vérité, avec une célérité merveilleuse »

After reading this first text he then read either the Encyclopédie or Duhamel du Monceau. The order is of no particular importance.

4.3.) Interpretation and use of the article « épingle (Pins) in the Encyclopédie

Once he had become interested in pin making Adam Smith consulted Diderot's Encyclopédie. He found Delaire's article, « Epingle ». The second paragraph spoke of 18 stages. Traduction 12 . The rest of the text was numbered 1 through 18. This would have no doubt reassured Smith that there were 18 stages in all.

At the end of the text he read: 18° put on the pins ……. traduction 13 ( No other production rate is given. Adam Smith probably believed this to be the production rate he was looking for, "the productive power of labour". If 36 000 pins "must" be papered each day, then the workshop has to produce 36 000. With 18 people working, one at each workstation, Adam Smith calculated the productivity rate at 2000 pins per day and per person.

4.4.) Interpretation and use of L'art de l'épinglier by Duhamel

Adam Smith acquired the text reviewed in the Journal des sçavans. He read it before, after or at the same time as the Encyclopédie. He read through Duhamel's Art of pin-making . He began by the introduction. During his readings he picked up the expression « division de ce travail » (division of this labour), so close to that used by Mandeville ("the dividing of labour"). He noted the number of stages now to be 17.

He saw the illustrations. The plates showed the tools, the machines, the labourers at work. The Académie des Sciences' illustrators, as those of the Encyclopédie, grouped the workers in one relatively large and tidy room. The labourers appeared to work in one workshop. Smith adopted this same view.

Smith used these three texts to document his Lectures on Jurisprudence. He would complete his documentation with the purchase of Macquer's Dictionnaire in France.

4.5.) Reading of the Dictionnaire portatif des arts et métiers by Macquer

In this new text, Adam Smith found understandable production rates. Macquer grouped together and simplified Perronet's figures that had previously been published by Duhamel. There are 8 distinct work rates. Smith would have been looking for a 12 « Une épingle éprouve dix-huit opérations avant d'entrer dans le commerce » 13 « On boute les épingles . C'est les placer dans le papier. On les prend à poignée, on les range par douzaine à la fois : il le faut bien, pour bouter jusqu'à 36 milliers d'épingles par jour ; encore ne gagne-t-on, quand on y excelle, que trois sous. » Was there any plagiarism? Given the extent of his personal contribution, it would seem not. The question is pertinent however as "he was peculiarly excitable about the idea of plagiarism" (Hamowy 1968, p. 253). Adam Smith's skill lay the way he used his different sources to construct an illustrative example of his thesis on the quantitative impact of the division of labour. He suggested the term "division of labour", a well devised term in light of previous scholarly work. The expression communicates both an idea of movement and that of ongoing specialisation. This dual meaning rendered the term a versatile one, and in so doing made it easier for others to work with the concept.

Adam Smith didn't use "productivity" to describe the second key concept of his theory. He called it "the productive power of labour". He could quite easily have used the more modern term "productivity". According to the Dictionnaire historique de la langue française the latter term was invented in France in 1766.

The multiple references hypothesis does not help explain one important part of Smith's pin making thesis. The production rate of 1 to 20 pins per day of a polyvalent worker is not given in the French texts. This reference still remains a mystery.

6) Bibliography

Bonar J., 1894, A catalogue of the library of Adam Smith, rééditions en 1932et 1966. Cannan E., 1896