Fabien Feschet
email: fabien.feschet@dg-medical.fr

The Saturated Subpaths Decomposition in Z 2 : a short note on generalized Tangential Cover

Keywords: Tangential Cover, digital curve, conservative predicates

In this short note, we generalized the Tangential Cover used in Digital Geometry in order to use very general geometric predicates. We present the required notions of saturated α-paths of a digital curve as well as conservative predicates which indeed cover nearly all geometric digital primitives published so far. The goal of this note is to prove that under a very general situation, the size of the Tangential Cover is linear with the number of points of the input curve. The computation complexity of the Tangential Cover depends on the complexity of incremental recognition of geometric predicates. Moreover, in the discussion, we show that our approach does not rely on connectivity of points as it might be though first.

Introduction

The Tangential Cover was first introduced as an algorithm in [START_REF] Feschet | Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature[END_REF] and fully recognized as a fundamental structure in subsequent publications [START_REF] Feschet | Canonical representations of discrete curves[END_REF][START_REF] Lachaud | Analysis and Comparative Evaluation of Discrete Tangent Estimators[END_REF]. Basically the Tangential Cover was the set of maximal Digital Straight Segments (DSS) which can be built on an input digital curve. Since its complexity is linear, computing the Tangential Cover is practically feasible and permits to tackle various problems on digital curves such as tangent computations or polygonalizations. The tangents obtained using the Tangential Cover does indeed converge towards their real counterpart for convex function. If the Tangential Cover was introduced using only DSS, it has also been used on extensions of DSS, such as thick segments [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF]. Moreover, thanks to the local notion of meaningful thickness [START_REF] Kerautret | Meaningful Thickness Detection on Polygonal Curve[END_REF], the Tangential Cover has been made adaptive allowing to perform analysis of noisy digital curves using automatic multi-scale selection [START_REF] Ngo | Analysis of Noisy Digital Contours with Adaptive Tangential Cover[END_REF]. It has been also included in the DGtal library [START_REF]DGtal: Digital Geometry tools and algorithms library[END_REF]. As it can be seen in the comments of the description of the source code of the DGtal library, extensions of the Tangential Cover is not straightforward in any case. In this paper, we explain what is a good notion of geometric predicates and proves that the Tangential Cover can be built with general predicates called conservative predicates still resulting in a linear output size complexity. We moreover make a careful clarification of the input of the Tangential Cover recalling that the Tangential Cover works for non simple curves as well as non connected sets of points. We hope that this short note will clarify the usage of the Tangential Cover.

Background notations

We consider, in this paper, points in the digital plane Z 2 and some adjacency α on those points. The usual adjacencies are the classical 4-and 8-adjacencies [START_REF] Klette | Digital Geometry -Geometric Methods for Picture Analysis[END_REF]. Let us recall that if we denote by N () a norm in R 2 , they are defined by the fact that two points p and q are adjacent if and only if N (q -p) ≤ 1. The norm N () is the l 1 norm -corresponding to the sum of the absolute differences of the coordinates of the points -for the 4-adjacency and it is the infinite norm l ∞corresponding to the max of the absolute differences of the coordinates of the points -for the 8-adjacency. In the sequel, the adjacency is fixed. Of course, our work readily extends to the case of other cellular decompositions of the real plane R 2 such as the triangular or the hexagonal decomposition but we restrict our presentation to the classical rectangular grid.

A digital α-path, shortly a digital path, in the plane is a list of points (p 0 , . . . , p n ) of Z 2 such that p i and p i+1 are α-adjacent and p i = p i+1 . It must be noted that a path is indeed an ordered set of points of Z 2 such that two consecutive points are α-adjacent. Hence, for a given set of points, they may be several paths describing it depending on the chosen ordering. A path is closed if and only if p 0 and p n are α-adjacent. When a path is closed, indices of the points in the path are intended modulo n + 1, the length of the path. For a path P , a subpath is a sublist (p i , . . . , p j ) of points of P which is an α-path. Care must be taken that α-connected subsets of P might not be subpaths (see Fig. 1) since we impose on each subpath to keep all points between p i and p j with respect to the ordering of the initial list describing the reference path.

For an α-path P , either closed or not, we denote by SP(P ) the set of all subpaths of P . This set contains O(n 2 ) elements by construction which is far less than the number (O(2 n )) of subsets of P .

A closed digital curve is a digital path where each point p i has exactly two α-adjacent points in the path. An open digital curve is a digital path with the same property except for the points p 0 and p n who have only one α-adjacent point in the path. These notions avoid the possibility of self-intersections and are, thus, too restrictive in practice. Hence, otherwise precisely mentioned, all digital sets manipulated in the paper will be digital paths.

Our main goal is to represent digital path as a graph structure. Since we only consider subpaths, they are always connected subsets by definition. So, we have a natural mapping ϕ between any α-path and a circular arc graph as follows. To do this, we fix an α-path P given by its list of points (p 0 , . . . , p n ). Every point in the path is sent via the mapping ϕ to a point on the unit circle Figure 1: An α-path with the order given by the arrows. The α-connected subpart depicted in bold line is not a subpath since the dotted part (in bold) is missing in the set. via the formula: ϕ(p k ) = exp(2iπk/(n + 1)) where i 2 = -1. This mapping can then be extended to map paths onto arcs of the circle. Indeed, any subpath of P is given by a list of consecutive points (p i , . . . , p j ) as shown on Fig. 2. The ϕ-image of the list is an arc given by (ϕ(p i ), ϕ(p j )). The ϕ mapping permits us to map any α-path onto a circular arc graph. This graph is an interval graph only when the path is open. It should be noted that ϕ is an inclusion preserving mapping that is, if S is a subpath of P then ϕ(S) is included in ϕ(P ).

As it can be seen in the definition of an α-path, the first problem to solve in our approach is to build an α-path from an image. Then, the ϕ mapping can be used to construct a graph from the path. Hence, in the next section, we explain in detail how to build our reference α-path from an image.

From images to path

The notion of α-path is intrinsically related to the notion of ordering of points. Indeed, we can define the canonical extension of an α-path P corresponding to the list of points (p 0 , . . . , p n ) by the piecewise linear function P can corresponding to the real polygon passing through the points from p 0 to p n and parametrized on [0, 1]. As each consecutive points are different, this clearly defines a function for which P can (0) = p 0 and P can (1) = p n . Hence, this function is continuous, pointwise and preserves the order between the real segment [0, 1] and the path P . As such, this function is a Jordan line [START_REF] Frechet | Les Espaces Abstraits[END_REF] (pp. 149-150) and [START_REF] Alexandrov | General theory of irregular curve[END_REF] (chapter 1). So, when starting from an image, the only problem to solve is to build an ordering of the points in the image as this completely defines a Jordan line. W.l.o.g., we can suppose that there is only one connected component with respect to the α-adjacency in the image because if it is not, all connected components can be treated separately.

We base our approach on the curve tracing algorithm with self-intersections considered by Hajdu and Pitas [START_REF] Hajdu | Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet[END_REF]. The key point of their approach is the notion of junction [START_REF] Klette | Digital Geometry -Geometric Methods for Picture Analysis[END_REF][START_REF] Klette | Branch Voxels and Junctions in 3D skeletons[END_REF]. We now briefly present their approach as well as its straightforward rewriting for our goal of constructing α-paths. It should be noted that as is, the method proposed by Hajdu and Pitas [START_REF] Hajdu | Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet[END_REF] is in fact somewhat imprecise as we will see later in this paper. This is mainly due to the fact that they do not consider complex junctions. We have corrected this little drawback in our version of their method and consequently still refer to it as Hajdu and Pitas' method.

The classification of simple curve is based on the notion of branching index. For a point p ∈ Z 2 , we consider V (p) as the set of points q ∈ Z 2 such that p and q are α-adjacent. V (p) is the α-neighbourhood of p. The branching index of p is then the number of points of q which also are image point. The classification of points in an image depending on their branching index [START_REF] Klette | Digital Geometry -Geometric Methods for Picture Analysis[END_REF] is the following:

• p is an end point if its branching index is 1, • p is a regular point if its branching index is 2,
• p is a branching point if its branching index is ≥ 3.

An α-junction is an α-connected subset of the image such that each point in the junction is a branching point and such that it is maximal for the inclusion order 1 . Hence, if a junction J 1 contains a junction J 2 then J 1 = J 2 . By extension, the branching index of a junction is the number of regular points which are α-adjacent to a point in the junction. If we consider a binary image I containing only one α-connected components and if we suppress all junctions in I, we get a simplified image Ĩ. This image might have several α-connected components but each connected component is a simple open curve. This process gives a decomposition of the image I into simple open curves and junctions. This naturally leads to a graph G = (V, E) as a representation of I as follows:

• V is composed of all end points and all junctions,

• E is the set of undirected edges {u, v} with u and v two vertices such that u and v are α-adjacent in I.

The key point in Hajdu and Pitas' algorithm is to remark that an α-path in the image I is thus a path in the graph G where each edge is used only once. Of course, this path might cross a junction more than once. So, the construction of an α-path is just the decomposition of the graph G into Eulerian subgraphs for which it has been proved long time ago that they always have an Eulerian tour, that is a path taking every edge exactly once. They also proposed a way of creating only one α-path for the whole connected component in I if repetition is allowed, that is some parts are used back and forth resulting in the multiple use of non branching points in the image (see part VI in [START_REF] Hajdu | Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet[END_REF]). This last problem was shown to be the Chinese Postman Problem, a well known problem in graph theory [START_REF] Schrijver | Combinatorial Optimization, Polyhedra and Efficiency[END_REF] (chap. 29 vol A). Originally Hajdu and Pitas' method is based on straight line segments and minimal length paths between two open curves sharing a junction. This is natural since their goals is also to apply a polygonal approximation algorithm on the image to efficiently compress it. However, two points are missing in their algorithm. First, they do not fully explain how they consider self-loop in the graph G. Indeed, self-loops cannot be distinguish in the graph G, hence they do not have any guarantee that their algorithm for constructing Eulerian tours will be able to take them in the right geometric order. This problem can have practical drawbacks but it is not a fundamental problem, to our point of view. The second problem with their method is that it is incorrect in full generality. Indeed, when the Eulerian tour as been constructed, they built an 8-connected path between pairs of open curves. However, the junction is not the union of the paths traversing it. Hence, only a subpart of the junction is coded by shortest paths between in and out points of a junction. It should be noticed that the JBEAM algorithm [START_REF] Huo | JBEAM: multiscale curve coding via beamlets[END_REF], cited in [START_REF] Hajdu | Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet[END_REF], does not have this problem. However, it is not suitable for our purpose since it does not consider paths but sets of points.

α-paths decomposition

The first step in our method consists in a decomposition of an α-path P into meaningful subparts, such as geometric primitives but not only. We must represent them in the most possible generic manner. For instance, several geometric primitives have been studied in the literature such as Digital Straight Segments (DSS), arcs of circles, arcs of parabolas and so on. More primitives can also be constructed and we shall derive a representation sufficiently generic for all such different basic elements. It appears that the most generic model is the model of predicates. A predicate, denoted by P, is a Boolean value function defined on SP(P ). We write P(X) if the predicate P has value true for the path X.

In the case of DSS, it has been proved that maximality is a basic property of digital segments on shapes [START_REF] Feschet | Canonical representations of discrete curves[END_REF][START_REF] Lachaud | Analysis and Comparative Evaluation of Discrete Tangent Estimators[END_REF]. We believe that if a subpath is meaningful, but is contained into another subpath also meaningful, we should retain only the largest one. Hence, we keep this principle of maximality in our construction. In the context of predicates, this notion was introduced by Mazurkiewicz [START_REF] Mazurkiewicz | Sur les lignes de Jordan[END_REF]. Definition 1. A set X is saturated with respect to a predicate P if 1. P(X) (X has the property corresponding to the predicate P),

2. every set Y ⊇ X with P(Y ) is such that Y = X.
In other word, a saturated set is a maximal set with respect to the inclusion order and for which the predicate is true. Clearly, when we consider saturated path, we also take into account the connectivity of the points since any path is connected. As a consequence, we can bound the number of saturated subpath for any predicate.

Lemma 1. Let P be a path with n points. Then, for any predicate P, there are at most O(n) saturated subpath in SP(P ).

Proof. We consider P as oriented positively with respect to the increasing order of the indices of its points. Let us consider a subpath X of P being composed of the sublist (p i , . . . , p i+k ) of points. We define its middle point as p m = p i+ k/2 . Then X is reconstructed by adding points around p m starting from the positive orientation. That is: p m , p m+1 , p m-1 , ... So to any subpath X, we can associate its middle point p m . This define a function m(X) = p m from SP(P ) to P . This function is obviously onto but not injective as symmetric subpath around p m all have the same middle point. But, given p m the subpaths of P having p m as middle points are totally ordered by inclusion. This is due to the fact that a subpath has no hole in the indices of its points. Hence, if we restrict the domain of m() onto the set of saturated subpaths then it becomes injective but loose its surjectivity. Indeed, for an arbitrary predicate it is possible that no sets M containing p m might be such that P(M ). For instance the predicate "to be a set containing p 0 " clearly has only one saturated subpath which contains p 0 . So since m() is injective, a point in P can not be used twice as a middle of a saturated subpath and since the function is not onto, not all points might be used. Hence, the number of points covered by the m() function restricted on saturated subpath is clearly bounded by n, hence the number of saturated subpaths is also bounded by n.

This property is false when applied to sets. For instance, the predicate "to have 3 points" has O(n 3 ) saturated sets but only O(n) saturated subpaths on P . It is important to notice that even if there are O(n 2 ) subpaths in P , only O(n) at most can be saturated. This implies that using saturated sets, algorithms with small, indeed linear if possible, time complexity are more easily reachable.

It is straightforward to see that for saturated subpaths, the inclusion is forbidden such that the corresponding graph is necessarily proper.

Property 1. The circular-arc graph constructed via the ϕ mapping of the saturated subpaths of any predicate P, is a proper circular-arc graph.

Let us now describe a simple algorithm which has been introduced in the context of maximal DSS [START_REF] Feschet | Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature[END_REF]. Our goal is to describe how to generalize this algorithm in the case of predicates. W.l.o.g., we suppose the path P to be closed. We start with an arbitrary point, for instance p 0 . Let us denote by S, the current tested set. The algorithm can be summarized by the following steps. 2. Check: if S is not a DSS then remove the last added point to S, else go back to the Increase step.

3. Maximality: Once S is symmetric and maximal then go back to the Increase step but only for one orientation. In case of ambiguity, the positive orientation must be chosen. When the Increase step stops again, move to the Restart step.

4.

Restart: S is a maximal DSS. Suppose that p m+r is its last point. Then let S + = S ∪ {p m+r+1 }. Remove points at the negative side of S + until it becomes a DSS. Then go to the Increase step to compute another maximal DSS.

This algorithm is proven to build the set of all maximal DSS of the path P in linear time [START_REF] Feschet | Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature[END_REF]. It is important to notice why the preceding algorithm is correct. Two facts are implicitly used, 1. If S is not a DSS then any set T containing S is not a DSS ; 2. Two consecutive maximal DSS shares at least one point.

The first property is used at the step Check of the algorithm to break the symmetry of S knowing that S cannot be extended any more on both sides and it is used at the step Maximality because when the one side extension of S has been built, it is declared to be a maximal DSS. The second property is used in the step Restart to ensure the fact that the decreasing of S always lead to a DSS and so the algorithm can start again.

Careful must be taken in the extension of this algorithm to predicates. Indeed, if we consider any predicate, the question of finding a subpath X such that P(X) probably could necessitate to test all subpaths leading to a quadratic complexity. Of course, this is practical for small n but our goal is to preserve the linear complexity of the algorithm during its extension, while keeping a sufficiently rich class of predicates containing at least classical geometric primitives.

We will restrict our attention to a special class of predicates in order to have interesting theoretical properties on the saturated subpaths. To do so, we only consider the notion of conservative predicate. Definition 2. A predicate P is conservative if and only if P(X) for a path X then for all Y ∈ SP(X), P(Y ).

In other words, when an conservative predicate is true on a path, it is true on all its subpaths. Of course, this also implies ab absurdo that if X is such that ¬P(X) then for any set Y containing X, we have ¬P(Y ). One important consequence is that if we know a set X for which the predicate is true or false then we can decide if there is a saturated set containing X. Hence, the simplest sets to test are the singletons. Indeed for any point p in P , if P({p}) then there exists a saturated set containing p ; else, it is clear that no saturated set contains the point p. This simple property is sufficient to modify the classical algorithm to obtain an algorithm valid for any conservative predicates.

It seems important to insist on the fact that the conservative property is a very natural property for geometric predicates. Indeed, if a path is a DSS, or an arc of a circle, or arc of a parabola then, it is natural to have that all subpaths are all DSS, arcs of circles or arcs of parabolas. The same is true for the negative case: if a set is not a DSS then clearly no superset can be a DSS ! The steps of the new algorithm are as follows.

1. Init: Let p be the current point. While ¬P({p}), go to the next point.

Stop either when all points in P have been tested or if a point p such that P({p}) has been found. In this case go to the Increase step.

2. Increase: this step is unmodified.

3. Check: this step is unmodified (simply use predicate P and not a DSS test).

4.

Maximality: This step is unmodified.

5.

Restart: S is a maximal subpath verifying P. Suppose that p m+r is its last point. Then let S + = S ∪ {p m+r+1 }. If ¬P({p m+r }) then set p = p m+r+1 and go to the Init step. Else remove points at the negative side of S + until it verifies P. Then go to the Increase step to compute another saturated subpath.

It is clear that this algorithm, referred as SSD (Saturated Subpaths Decomposition), can be improved by a test on all singletons. Indeed, each time a point p is used, if ¬P({p}) then no saturated set containing p can exist such that we can directly move to the next point in P . This optimization does not change however the complexity of the algorithm. For the same reason than in [START_REF] Feschet | Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature[END_REF], this new version of the classical algorithm still has a linear time complexity in terms of the number of time the predicate is checked. Hence, for constant time predicate, this extension still runs in linear time.

Theorem 1. The algorithm SSD is correct, that is, it computes all saturated subpaths of a conservative predicate P.

Proof. Let us suppose w.l.o.g. that P({p 0 }). There are two cases: either there exists a saturated subpath S 0 such that m(S 0 ) = p 0 , or there is a saturated subpath S 0 which contains p 0 but not as a middle. In the second case, several such subpaths might exist and S 0 is chosen to be the one with the rightmost middle. In the first case, the Increase and Check steps are built to find S 0 so the algorithm is correct if and only if it correctly moves to the next saturated subpath.

In the second case, the Increase-Check steps build a set S whose middle is p 0 such that P(S). Then the Maximality step is used. Suppose that S has the form (p i , . . . , p j ). There are two cases : either P({p i , . . . , p j+1 }) or ¬P({p i , . . . , p j+1 }). In the first case, the Maximality step implies that the Increase step is pursued on the positive side and in the second case, on the negative side. On both sides, a saturated set is built with a rightmost middle, that is S 0 . Hence, once again, the algorithm is correct if and only if it correctly moves to the next saturated subpath.

So the key point is the Restart step. Let S = (p i , . . . , p j ) be the last computed saturated subpath. Either P({p j+1 }) or ¬P({p j+1 }). In the first case, there exists a saturated set containing p j+1 . Since this point was not in S, it is clear that its middle is at the right of m(S). Hence the Restart step stops necessarily at the left of this middle point and then the Increase-Check steps are used to find the correct middle point.

In the second case, no saturated subpaths contain p j+1 thus, the Restart step move to p j+2 and the Init step is used to find a new point belonging to a saturated subpath. So, no saturated subpath can be missed and thus the algorithm is correct provided than it stops when all points have been tested.

Discussion

In the proof of Theorem 1, we mainly used the middle point of a subpath. This is the common way to uniquely identified a saturated subpath. However, it is possible, in practice, to use only forward incremental recognition. Indeed, this leads to the same Tangential Cover with the only careful check of the first computed primitive. Starting at any point of the curve does not imply that this point is a beginning point of a saturated subpath. Hence, to complete the construction of the Tangential Cover, this first computed primitives should be tested for maximality and suppressed if it is not maximal. This means that the algorithm must be pursued until the end of the first recognized primitive is reached. Obviously the complexity remains the same.

We must also remark that our proof does not rely on the adjacency α. Indeed, we only use the indices of the points. Hence, an adjacency can be based solely on a numbering of the points. Hence, the points does not need to be connected in the usual sense, they are connected by indices only. Of course, we should add that a good geometric predicate must be true on any singletons. Giving a numbering is a more profound property than connectivity in fact. The only case to forbid is the repetition of points which consecutive indices. This might be a problem for computing the saturation of subpaths. Indeed, at first saturation of paths might be viewed as a set maximality for inclusion, but it is more general to have a maximality with respect to connected subsets of the set of indices. In that case, maximality does not imply that the points in paths are different between saturated sets but this avoid useless complexity in the decomposition.

So to conclude, the original set of points can described very general curves, allowing non simplicity and non connectivity. This is not a problem since the canonical extension P can is properly defined on both cases.

Conclusion

In this short note, we clarified the generalization of the Tangential Cover to arbitrary geometric predicates using the usual notion of conservatism. This work was done in 2009 but never published. When reading the source code of the DGtal library [START_REF]DGtal: Digital Geometry tools and algorithms library[END_REF], I have discovered that concepts of segments (CForwardSegmentComputer) where supposed to be true in specific subsets of a segment. Hence, I have decided to write this note to prove that the size complexity of this generalized Tangential Cover is still linear.

Figure 2 :

 2 Figure 2: (left) An α-path with the order written inside the pixels (right) The mapping of the points and the overlapping (31,37) and (36,45) subpaths.

1 .

 1 Increase: add points to S one by one, always starting in the positive orientation. Hence, if S is a subpath of the form (p m-k , . . . , p m+k ) then first add p m+k+1 before p m-k-1 . If S is of the form (p m-k+1 , . . . , p m+k ), then add p m-k .

The maximality with respect to inclusion is essential in the definition of a junction but is missing in[START_REF] Hajdu | Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet[END_REF] and in[START_REF] Klette | Branch Voxels and Junctions in 3D skeletons[END_REF] even if it is evident from their papers that they both do consider maximal junctions.


Keywords: Scientific thought experiments, Galileo's falling bodies, De Motu (1590) and Discorsi

function and limit of Galileo's falling bodies thought experiment: Absolute weight, Specific weight and the Medium's resistance

Introduction

 Galileo's Discorsi (1638)falling bodies TE has become a key case study in the epistemological literature on TEs, especially since [START_REF] Brown | Thought Experiments since the Scientific Revolution[END_REF] famously claimed that it is canonical case of what he labelled "platonic TE": it is both destructive and constructive. It is destructive since it refutes an old theory (i.e. Aristotle's theory of free-fall), it is also constructive since it establishes, in a priori fashion, a new law of nature (i.e. in void, all bodies free-fall at the same speed). Brown's analysis was met by [START_REF] Norton | Are Thought Experiments Just What You Thought?[END_REF] who denied this "platonic" power of the TE and argued that it is reducible to a deductive argument, a TE-argument, that is an argument with irrelevant and even eliminable particulars. Both Norton and Brown agree that the TE perfectly leads to its destructive conclusion in a deductive manner. In addition, if the TE leads to its constructive conclusion, Norton claims that the TE-argument could deductively lead to this conclusion as well. However, Norton argues that the TE-argument shows us that the TE only leads to its constructive conclusion if we add the following hidden assumption 8a: the speed of a falling body depends only on its weight (see 4). Which for Norton amounts to assuming vacuum, something Galileo could not do in the context of the TE, and thus this constructive conclusion is "at worst, a fallacious inference to a falsehood [when assumption 8a does not hold]; or, at best, valid only insofar as it is invoked in special cases in which assumption 8a holds, such as the fall of very heavy, compact objects in very rare media. This final step now looks more like a clumsy fudge or a stumble than a leap into the Platonic world of laws." (Norton 1996: 345, my emphasis).

Norton's concluding remark, apart from being in tension with his "elimination thesis" 1 , since he seems to grant some important role for the particulars involved it this TE, elicit the need to analyse the function of the particulars involved in Galileo's TE: very heavy, compact spherical objects of the same material falling in a rare medium such as air. More generally, the literature on TEs suffers from a major omission in analysing this TE: it does not tackle Galileo's restriction, in the TE's scenario and conclusion, to bodies of the same material. This omission is not proper to the Norton/Brown debate but is found in most of the literature. Of course, we find here and there some mention. For instance, Gendler underlines in a footnote the "somewhat unfortunate practice of considering this thought experiment outside of both its historical and textual contexts" (Gendler 1998: 402, ft 8). She then briefly mentions this restriction, however without analysing it, since she believes that for the purpose of her discussion "this constraint is irrelevant" (403, ft 13). Even if she rightly concludes that the TE's function is refutational and claims that she doesn't "think that the thought experiment in question shows anything more than that natural speed is independent of weight" (419), this restriction should not be left unanalysed if we want to understand the function and limit of Galileo's TE and its role in both argumentative strategies.

This restriction has even puzzled many Galilean scholars. For instance, at the end of his historical analysis of Galileo's arguments and TEs ("experiences imaginaires") since the De Motu (1590), Alexandre Koyré states that "Galileo's mention of specific gravityand this, in a reasoning in which it has nothing to dois extremely curious. And even, historically, very important." 2 . More recently, [START_REF] Palmieri | Spuntur lo scoglio più duro': did Galileo ever think the most beautiful thought experiment in the history of science?[END_REF] and [START_REF] Van Dyck | An Archeology of Galileo's science of motion[END_REF] analysed the historical development of this TE and its restriction to bodies of the same material, which brought Palmieri to conclude that "[p]erhaps we need a new approach to the question of thought experiment, capable of integrating results from different disciplinary areas, such as, for instance, the history and philosophy of science and cognitive science. […] The all too clean baby of today's debate on the most beautiful thought experiment in the history of science [Galileo's TE] should definitely be thrown out, and the bathwater carefully analyzed." (Palmieri 2005: 238) Regrettably, this was not taken into account by most philosophers working on TEs. For instance, we still find in the Stanford entry on TEs (2017, substantively updated in 2014) and in Brown's second version of his book (2010), that "Galileo showed that all bodies fall at the same speed with a brilliant thought experiment" (my emphasis).

I am in total agreement with Palmieri that history of science should play a central, at least a much greater role in the philosophical debate on TEs. Indeed, we have the general impression that the epistemological literature on scientific TEs is mainly built on a-historical analysis of case studies. This is especially lamentable for Galileo's falling bodies because the epistemological literature takes this TE as a canonical case study, while the a-historical analysis of this TE yields wide disagreements about its conclusion(s), leading to divergences pertaining to its epistemic function. Thus, leading the epistemological literature on TE astray and turning an important debate into a red herring: the Norton/Brown debate on TEs revolves, in part, around how Galileo's TE justifies its conclusions, by direct a priori access to laws of nature or by being a deductive argument. Nevertheless, the TE's function is misrepresented as revealing and justifying a law of nature (Brown since 1986 and even in a sense in Norton's 1996 reply).

The philosophical literature is thus in need of a more careful historical analysis of Galileo's TE and the following questions answered, before trying to analyse if and how the TE could justify its conclusion(s): What is the TE's function (or intended conclusion) for Galileo?

1 "Thought experiments are arguments which contain particulars irrelevant to the generality of the conclusion. Thus any conclusion reached by a good thought experiment will also be demonstrable by an argument which does not contain these particulars and therefore is not a thought experiment" (Norton 1991: 131). 2 "La mention par Galilée de la gravité spécifiqueet ce, dans un raisonnement où elle n'a que faireest extrêmement curieuse. Et même, historiquement, très importante." my emphasis and translation, Koyré 1960: 203. What is its role in Galileo (1590 and 1638)'s argumentative strategies? What is the function of the particulars involved in its scenario? What are the idealisations involved? Are these idealisations justified? Since vacuum could not be explicitly assumed in the TE and thus its scenario takes place in plenum, then how did Galileo take into account the multiple effects of the medium's resistance? In case we assume vacuum (for modern readers), what conclusion could the TE lead to? Is this conclusion justified? All these questions could be easily answered once we tackle the more general one: why is the TE restricted to bodies of the same material?

This paper aims at analysing the function and limit of Galileo's falling bodies TE, which will provide answers to these questions. First in (2), I show that the TE's function is only refutational; it aims at refuting Aristotle's theory of free-fall, one of its two principles to be precise, by showing that the falling body's absolute weight could not cause divergences in the speed of free-falling bodies. I thus retrace Galileo's TE to its first occurrence in the De Motu (1590) which explicitly indicates Galileo's intention of "seeking causes of effects". Second in (3), I analyse Galileo's both argumentative strategies that led him to two incompatible theories of free-fall. It will be shown that the TE's restriction to bodies of the same material is best understood when placing the TE in both 1590 and 1638 argumentative strategies. I will argue that both strategies aimed at exploring potential causal factors affecting divergences in speed of free-falling bodies: the TE aimed at eliminating absolute weight as a causal factor, which explains Galileo's restriction to bodies of the same material, while both 1590 and 1638 subsequent arguments aimed at exploring specific weight as a causal factor, with conflicting conclusions. Third in (4), I analyse one small effect of the medium's resistance that could not be taken into account in the TE, even by Galileo's choice of particulars; i.e. the medium's disproportionate effect on the free-falling body's surface to absolute weight ratio. This shows that the TE only works either if we can assume vacuum or by placing the TE in the whole argumentative strategy, where this small effect of the medium's resistance is subsequently explained (which Galileo does in 1638) and thus could be ignored in the TE. Finally in (5), I summarize, draw out two implications for the debate on TEs and restate answers to the above questions.

Absolute weight in the De Motu (1590) and the Discorsi (1638): same TE, same conclusion

Galileo3 first introduced his TE in the De Motu, an unpublished manuscript usually dated from the 1590's. The TE appears in a larger argumentative strategy intended to first refute Aristotle's theory of free-fall and then defend Galileo's own early theory.

Galileo starts by clarifying the concepts of "heaviness" and "lightness". He stresses that both should be understood by what we could call "specific weight" (even if Galileo is comparing equal volumes, not unit volumes of bodies), without explicitly defining the concept in the De Motu. Indeed, Galileo tells us that "a thing should be called heavier than another, if when a piece of it is taken, equal to a piece of the other, it is found to be heavier than the piece of the other" (Galileo 1590: 1).

Then Galileo distinguishes different ways in which "greater or lesser swiftness of [natural] motion comes about" (Galileo 1590: 14). This is best understood when we divide, following Galileo, Aristotle's theory of free-fall into two principles 4 : (i) natural speed is proportional to weight, (ii) natural speed is inversely proportional to the medium's resistance or "density":

[I]nequalities in the slowness and swiftness of motion occur in two ways: for either the same mobile is moved in different media [i.e. according to Aristotle's principle (ii), the speed of a mobile is inversely proportional to the medium's resistance]; or the medium is the same, but the mobiles are different [i.e. according to Aristotle's principle (i), the speed of the mobile is proportional to its weight]. We will demonstrate shortly that in both cases of motion the slowness and swiftness depend on the same cause, namely, the greater or lesser heaviness [i.e. specific weight] of the media and of the mobiles; but first we will show that the cause of such an effect which has been conveyed by Aristotle is insufficient. (Galileo 1590: 14, my emphasis 5 ) Galileo's aim is to be found in this passage: he is seeking the causes of inequalities of slowness and swiftness of motion. He first aims at showing that the causes conveyed by Aristotle are either falseprinciple (i)or insufficientprinciple (ii). Galileo then aims to propose an early theory of free-fall, according to which the speed of a free-falling body is proportional to its specific weight (to be precise, minus the specific weight of the medium, see 3.1). This is how he proceeds.

Galileo starts by arguing against principle (ii) which states "that the cause of the slowness of motion is the thickness of the medium, and that of the speed, its subtlety" (p.14). Galileo aims at showing that this cause is insufficient, and he demonstrates this by appealing to examples where bodies, such as an inflated bladder, fall slowly downwards in air, but fly very swiftly upward when let go from deep in water.

Then Galileo moves to principle (i), which is the purpose of the TE. This principle aims at describing the speed of mobiles falling in the same medium. For these mobiles, Galileo further distinguish between two cases:

[D]ifference between two mobiles can happen in two ways: for either they are of the same species, as, for example, both lead, or both iron; and they differ in size: or they are of different species, e.g. one iron, the other wood; they then differ from one another either in size and heaviness, or in heaviness and not in size, or in size and not in heaviness. (Galileo 1590: 15) This distinction is crucial for what follows. Galileo will first limit his arguments against Aristotle to the first case; to bodies of the same species that differ only in size. For these bodies the difference in size is directly translated into a difference in absolute weight and mostnot all (see 4) -of the effects of the medium's resistance are the same. While for bodies of different species things are more complicated; they could differ in the three different ways enumerated above (see Fig. 1). After his TE, Galileo will analyse bodies of different specific weights and bodies falling in different media simultaneously (see 3.1).

same ratio as the weights; so that, for example, a body which is ten times as heavy as another will move ten times as rapidly as the other. In the second case he assumes that the speeds of one and the same body moving in different media are in inverse ratio to the densities of these media; thus, for instance, if the density of water were ten times that of air, the speed in air would be ten times greater than in water. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF]" Galileo /1914: [105-106] : [105-106] of the National Edition. 5 All emphasis in the subsequent quotes from the De Motu and the Discorsi are mine. For bodies of the same species differing only in size (bottom left of Fig. 1), Galileo starts by explicitly stating Aristotle's principle (i):

Concerning those mobiles that are of the same species Aristotle has said, that the larger is moved faster [...] Aristotle wants mobiles of the same genus to observe between themselves in the speed of motion the ratio of the sizes that these mobiles have: and he says that very openly […], by affirming that a large piece of gold is carried more swiftly than a small one. (Galileo 1590: 15-16) For bodies of the same species, principle (i) amounts to saying that the speed of a freefalling body is proportional to its volume or absolute weight. Galileo first dismisses this principle on empirical, or semi-empirical observations:

How ridiculous this opinion is, is clearer than daylight: for who will ever believe that if, for example, [...] from a high tower, two stones, one being double the size of the other, were thrown at the same moment, that, when the smaller was at mid-tower, the larger would already have reached the ground?" (Galileo 1590: 16) Then Galileo moves away from empirical examples to several arguments, the last one being his famous TE. Galileo starts by explaining his preference to appealing to non-empirical arguments in "seeking the causes of effects":

[I]n order that we may always make more use of reasons than of examples (for we are seeking the causes of effects, which are not reported by experience), we will bring forth our way of thinking, whose confirmation will result in the downfall of Aristotle's opinion. (Galileo 1590: 16) Galileo's entire thought process is somehow nested in two Archimedean analogies. The first concerns bodies floating on water, while the second concerns bodies heavier than water and thus sinking (see 3.1). Concerning the first Archimedean analogy, Galileo claims that the reason why bodies of the same species fall at the same speed is analogous to why a large beam and a small piece of wood float on water:

We say, then, that mobiles of the same species (let those things be said to be of the same species that are constituted of the same material, such as lead or wood, etc.), though they may differ in size, are however moved with the same swiftness, and a larger stone does not go down more swiftly than a smaller one. Those who are surprised by this conclusion will also be surprised that a very large beam can float on water, just as well as a small piece of wood: for the reasoning is the same. (Galileo 1590: 16-17) Before introducing his TE, Galileo first proposes a three steps argument6 to explore this Archimedean analogy against Aristotle's principle (i):

In the first, Galileo invites us to think about the behaviour of a wooden beam and a stick of the same wood floating on the surface of the water. Galileo asks to imagine that the water's specific weight decreases to the point that it becomes lighter than the wood's. Then he asks, "who would ever say that the beam would go down first or more swiftly than the small piece of wood?" (Galileo 1590: 17).

In the second stage, Galileo reverses the strategy of the first. Instead of the medium's specific weight decreases, now the body's specific weight increases. He asks to imagine a volume of wax that is gradually filled with sand until the mixture's specific weight becomes bigger than the water's. Galileo then asks: "who would ever believe, if we took a particle of such wax, say one hundredth of it, either that it would not go down or that it would go down a hundred times more slowly than the totality of the wax?" (Galileo 1590: 17) These analogies show Galileo's emphasis on specific weight rather than absolute weight when analysing the speed of free-falling bodies. This is especially reflected in the third stage, where he explores the analogy between a balance and bodies floating on water: And it will be possible to experience the same thing in the balance: for if very large, equal weights are placed on each side, and then to one of them something heavy, but only modestly so, is added, the heavier will then go down, but not any more swiftly than if the weights had been small. And the same reasoning holds in water: for the beam corresponds to one of the weights of the balance, while the other weight is represented by an amount of water as great in size as the size of the beam: if this amount of water weighs the same as the beam, then the beam will not go down; if the beam is made slightly heavier in such a way that it goes down, it will not go down more swiftly than a small piece of the same wood, which weighed the same as an [equally] small part of the water, and then was made slightly heavier. (Galileo 1590: 17) This third step can be interpreted as an exploration of the analogy between the role of absolute weight on the balance7 and the role of specific weight in the floatability of bodies on water:

In the first case, the equilibrium of the balance is broken if one adds a weight on one arm of the balance in equilibrium. Whatever the material of these two weights or the added weight, what matters is the difference between the absolute weight that is already on the scale and the added weight. The mobile "falls", so to speak, on an arm of the balance when an extra weight is added. This speed of fall does not depend on the initial body's absolute weight, but on the difference between the absolute weight of the initial body and that of the added body.

In the second case when analysing bodies sinking in water, this difference must be understood in terms of specific weight. It is when one changes the body's specific weight that the equilibrium, which existed between the body and the water, is broken, and the floating body then sinks. It sinks with the same speed, whatever its volume or absolute weight, a beam or a stick of wood. The speed of fall does not depend on the initial body's absolute weight, but on the difference between the specific weight of the floating body and the specific weight of the added body. Galileo will indeed defend at the end of his argumentative strategy that the speed of a free-falling body is proportional to the specific weight difference between the mobile and the medium (see 3.1). But first, Galileo will argue against Aristotle's principle (i) with his TE.

2.1

The TE in the De Motu Galileo introduces his TE as follows "[b]ut it is pleasing to confirm this by another argument". Its scenario is almost the same as that of the Discorsi, five decades later. The difference is in Galileo's justification of the following mediativity principle: And first, let the following be presupposed: namely, if there are two mobiles, one of which is moved faster than the other, the combination of the two is moved more slowly than that part which was moved faster than the other, but more swiftly than the remaining part, which, alone, was carried more slowly than the other. (Galileo 1590: 17-18) In the Discorsi, Galileo justifies this supposition with the following theoretical axiom "each falling body acquires a definite speed fixed by nature, a velocity which cannot be increased or diminished except by the use of force" [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [107]: [107]). In the De Motu, this supposition is justified by appealing to two examples taken from empirical observations: the first concerns two mobiles ascending in water 8 , while the second concerns two mobiles of different material falling in air:

[I]f [...] two mobiles go down, one of which is carried more slowly than the other, as, for example, if one is wood, the other a bladder, which go down in air, the wood more swiftly than the bladder, we presuppose this: if they are combined, the combination will go down more slowly than the wood alone, but more swiftly than the bladder alone. For it is manifest that the swiftness of the wood will be retarded by the slowness of the bladder, while the slowness of the bladder will be accelerated by the speed of the wood; and similarly a certain measure its weight. cf. Van Dyck 2006 for the analysis of the evolution of the role of the balance in Galileo's reasoning on free-fall. 8 "As, for example, if we understand two mobiles, such as a piece of wax and an inflated bladder, both of which are carried upward from deep water, but the wax more slowly than the bladder, we ask that it be conceded, that if they are combined, the combination will go up more slowly than the bladder alone, but more swiftly than the wax alone. Indeed this is very clear: for who doubts that the slowness of the wax will be diminished by the speed of the bladder, and, on the other hand, that the speed of the bladder will be retarded by the slowness of the wax, and that a certain motion intermediate between the slowness of the wax and the speed of the bladder will result?" (Galileo 1590: 18). motion intermediate between the slowness of the bladder and the swiftness of the wood will result. (Galileo 1590: 18) Note that this justification may seem to be out of place in the context of the TE, since Galileo considers, as in the version in the Discorsi, two bodies of the same material. Nevertheless, this mediativity supposition is not weakened: Galileo, through these examples, seems to give it an empirical, or semi-empirical justification resulting from our daily experience.

Galileo then aims, with the following TE's scenario, at showing an inconsistency between this mediativity principle and Aristotle's principle (i):

Let there be two mobiles of the same species, the larger a, and the smaller b; and, if it can be done, as our adversaries hold, let a be moved more swiftly than b. There are then two mobiles one of which is moved more swiftly than the other; hence, according to what has been presupposed, the combination of the two will be moved more slowly than a alone: but the combination of a and b is larger than a alone: hence, contrary to our adversaries' view, the larger mobile will be moved more slowly than the smaller; which would certainly be unsuitable. (Galileo 1590: 18) That is in unfolding9 the TE's scenario according to these two principles, we arrive at an absurd result describing the composite body falling, at the same time, both faster and slower than the larger body a. Which brings Galileo to conclude: Accordingly, let it be sufficiently confirmed that there exists no cause, per se, why mobiles of the same species should be moved with unequal speeds but there certainly is one why they should be moved with equal speed. But if there were some accidental cause, such as, for example, the shape of the mobile, it must not be classified amongst the causes per se. (Galileo 1590: 18) This I submit is the function of the TE which is reflected in Galileo's own words in the De Motu: Galileo is isolating absolute weight in order to analyse it as a potential factor that could cause divergence in speed of free-falling bodies. He concludes, from his TE, that absolute weight could not be a causal factor and thus, contrary to Aristotle's principle (i), bodies of the same material do not fall proportionally to their absolute weight. Thus, bodies of the same material will fall with the same speed, if all "accidental" causes are accounted for. However, the TE remains silent concerning the effects of other causal factors, in particular specific weight.

I add that the TE's function will remain the same in the Discorsi (see 4), 5 decades later. The difference between these two occurrences of the same TE is to be found in the subsequent arguments that aimed at exploring specific weight as a potential causal factor, with two conflicting conclusions. To see that, let us compare how Galileo defended two incompatible theories of free-fall with two different argumentative strategies in 1590 and then in 1638.

Specific weight in the De Motu (1590) and the Discorsi (1638): two argumentative strategies, two theories of free-fall

In this section I aim at showing how Galileo defended two different theories of free-fall with different arguments that followed the same TE. I will show that this difference could be traced to Galileo's treatment of specific weight as a causal factor. I will first expose Galileo's De Motu second Archimedean analogy which led him to defend his early theory of free-fall: in void all bodies fall with a speed proportional to their specific weights. Second, I will expose how in the Discorsi, Galileo eliminated specific weight as a causal factor to defend his final theory of freefall: in void, all bodies of any material, fall at the same speed.

De Motu's Archimedean analogy: specific weight is a causal factor

The De Motu provides a very interesting manuscript to understand the evolution of Galileo's thought process, the limited function of his TE and his struggle with the causal role of specific weight, especially when compared with the Discorsi's argumentative strategy. In the De Motu, having eliminated absolute weight as a causal factor with the TE, by restricting its scenario to bodies of the same material, Galileo now wants to show that for bodies of different species, Aristotle's principle (i) is also false. First, Galileoby building on the equality of speed for mobiles of the same species differing only in sizereduces his analysis of mobiles of different species, which differ in the three ways listed in Fig. 1, only to those differing "in heaviness and not in size". He argues that "if the ratio of the motions of those mobiles that differ only in heaviness and not in size is given, the ratios of those that differ in any other way are also given" (Galileo 1590: 19). Then Galileo tackles both principles simultaneously: And so, in order that we may find this ratio and, against Aristotle's way of thinking, show that in no way do mobiles observe the ratio of their heavinesses, even if they are of different species, [i.e. principle (i)] we will demonstrate things on which depends the answer not only to this investigation, but also to the investigation of the ratio of the motions of the same mobile in different media [i.e. principle (ii)]; and we will examine both questions simultaneously. (Galileo 1590: 19) Galileo will examine both principle simultaneously with his second Archimedean analogy. Recall the first (see 2) Archimedean analogy concerned a large beam and a small piece of wood floating on water and Galileo used it to refute principle (i) for bodies of the same material. Galileo will build on the following second analogy to refute both principles and to defend his early theory of free-fall:

[A]ll these things will easily be drawn from the following demonstration. I say, then, that a solid magnitude heavier than water is carried downward with as much force as that by which a quantity of water, having a size equal to the size of the same magnitude, is lighter than this magnitude." (Galileo 1590: 23) Galileo provides several proofs of this latter claim (cf. Palmieri 2005: 229) and then concludes following this Archimedean analogy and contrary to principle (ii) that:

[T]he same mobile going down in different media, observes in the swiftness of its motions, the ratio to one another of the excesses of its own heaviness over the heavinesses of the media: thus if the heaviness of the mobile is 8, but the heaviness of a size of one medium, equal to that of the mobile, is 6, then the swiftness of this body will be 2; if the heaviness of an amount of the other medium, equal to the size of the mobile, is 4, then the swiftness of the mobile, in this medium, will be 4. It is therefore evident that these swiftnesses will be to one another as 2 and 4; and not as the thicknesses or the heavinesses of the media, which is what Aristotle wanted, which are to one another as 6 and 4 (Galileo 1590: 24) Galileo then applies the same reasoning to the fall of different bodies in the same medium, and concludes contrary to principle (i):

Similarly the answer to the other question is evident: namely, what ratio the speeds of mobiles equal in size, but unequal in heaviness, observe with one another in the same medium. For the speeds of such mobiles will be to one another as the excesses by which the heavinesses of the mobiles exceed the heaviness of the medium: thus, for example, if two mobiles are equal in size, but unequal in heaviness, the heaviness of one being 8, and of the other 6, but the heaviness of an amount of the medium, equal in size to the size of one of the two mobiles, is 4, then the swiftness of the former mobile will be 4, and that of the latter will be 2. Hence these speeds will observe the ratio of 4 to 2; and not that which is between the heavinesses, namely 8 to 6. (Galileo 1590: 24) Put differently, Galileo's early theory describes the speed of free-falling bodies as an Archimedean ratio (a subtraction), not geometric (a division) as Aristotle's wanted: The speed is not W/R (Weight/Resistance), but proportional to Wb -Wm (Wb and Wm being the specific weights of the body and the medium respectively)10 . Thus, according to this early theory, in void where Wm = 0, mobiles fall proportionally to their specific weight: Thus, in a void also a mobile will be moved in the same way as in a plenum. For in a plenum a mobile is moved swiftly according to the excess of its own heaviness over the heaviness of the medium through which it is moved; and thus in a void it will be moved according to the excess of its heaviness over the heaviness of the void: since this is null, the excess of the heaviness of the mobile over the heaviness of the void will be the total heaviness of this same mobile; thus it will be moved swiftly according to its own total heaviness. (Galileo 1590: 32) That is in the De Motu, Galileo defends that specific weight is a causal factor affecting speed, even in void. In addition, this theory is consistent with the TE since specific weight is a mediative property. That is combining in the TE's scenario two bodies of different specific weights results in a body whose specific weight lies between the specific weights of the two constituent bodies. Which means that -according to the mediativity principle, but also according to the De Motu's theory -the combined body should fall at an intermediate speed.

To see this, consider Gendler's reconstructed argument which aims at revealing a contradiction between 3 premises; i.e. (1) speed is mediative, (2) absolute weight is additive (3) natural speed is directly proportional to absolute weight. This shows a contradiction since a "mediative property cannot be directly proportional to one that is additive" (Gendler 1998, p.404). But we can rewrite the argument as follows without any contradiction: (1) speed is mediative, (2) specific weight is mediative (3) natural speed is directly proportional to specific weight. As we will see (3.2), in the Discorsi Galileo needed additional arguments to eliminate specific weight as a causal factor.

Finally, it should be noted that Galileo, immediately after defending the Archimedean ratios in plenum, notes that "a very great difficulty arises here: it will be found that these ratios are not observed by one who has made a test." However, without exploring this further, since he is convinced that "[i]t is necessary first to examine certain things which have not yet been inspected. For it is necessary, first, to see why natural motion is slower at the beginning." (Galileo 1590: 24) 3.2 Discorsi's limiting case argument: specific weight is probably not a causal factor Five decades later, the same TE was reused by Galileo for the same purpose. However, the TE now constituted the first step of a more complex argumentative strategy which spans for 30 pages. Following the TE, Galileo will now propose two additional arguments, which will bring him to eliminate specific weight as a causal factor and to defend his final theory of free-fall: in void, all bodies fall at the same speed. This is how he argued. Following the TE, the second step consisted of a limiting case argument: SALV. [...] in a medium of quicksilver, gold not merely sinks to the bottom more rapidly than lead but it is the only substance that will descend at all; all other metals and stones rise to the surface and float. On the other hand the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [116]: [116]) It is thus this "observation", not the TE, that brought Galileo to the conclusion that in void, all bodies would fall at the same speed. Having observed that the variation of speed, of bodies of different specific weights, becomes less and less important with the ratification of the medium, we could extrapolate what will happen at the limit in a medium totally devoid of resistance:

[...] if we find as a fact that the variation of speed among bodies of different specific gravities is less and less according as the medium becomes more and more yielding, and if finally in a medium of extreme tenuity, though not a perfect vacuum, we find that, in spite of great diversity of specific gravity [peso], the difference in speed is very small and almost inappreciable, then we are justified in believing it highly probable that in a vacuum all bodies would fall with the same speed. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [117]: [117]) But, as it is made explicit by Galileo, we are justified in believing that in void all bodies fall at the same speed, only as "highly probable"11 . In fact, this is the case since this limiting case argument is also consistent with the De Motu early theory: at the limit, and therefore in a vacuum, the differences in speed between two falling bodies with different specific weights could only be small, not null. As shown in Palmieri's diagrams (Fig. 2), both Galilean theories predict divergences in the speed of falling bodies of different materials in plenum. In addition, both theories are consistent with this limiting case argument since they also predict that this divergence decreases with the ratification of the medium. However, in void, the two theories give two different predictions: in the diagram to the left, with the "restricted" De Motu's theory, when the resistance of the medium becomes null, there will always be a small difference in the speed of falling bodies of different material. In void, the sphere made of gold falls faster than the one made of gold + silver, since specific weight is a causal factor according to the De Motu's theory. While in the diagram on the right, with the "general" Discorsi theory, the speed of all falling bodies will be identical in void, since specific weight is no longer a causal factor. Fig. 2: Inequalities in speed according to both theories (Palmieri 2005: 234) Galileo needed one additional step in his argumentative strategy in order to pass from "highly probable" to "confirming" his theory of free-fall. Which will provide him with a way to choose between his early and final conflicting theories, that is to make a theoretical choice. Galileo will provide an argument which aims at eliminating specific weight as a causal factor affecting the speed of free-falling bodies in void.

Discorsi's constant cause, constant effect argument or fall from small and high altitudes: specific weight is not a causal factor:

Galileo starts by setting up the stage for his analysis of different bodies falling from different altitudes. Salviati first raises and answers the following question: SALV. [...] Now, Simplicio, if we allow these two bodies [an inflated bladder and a mass of lead having the same size] to fall from a height of four or six cubits, by what distance do you imagine the lead will anticipate the bladder? You may be sure that the lead will not travel three times, or even twice, as swiftly as the bladder, although you would have made it move a thousand times as rapidly. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [117]: [117])

To which Simplicio agrees, but adds that if they fall from a high altitude the difference will be bigger: SIMP. It may be as you say during the first four or six cubits of the fall; but after the motion has continued a long while, I believe that the lead will have left the bladder behind not only six out of twelve parts of the distance but even eight or ten. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [117]: [117]) Which will provide Salviati with the opportunity to analyse this divergence in speed of fall from different altitudes, all the while confirming that specific weight could not be a causal factor:

SALV. I quite agree with you and doubt not that, in very long distances, the lead might cover one hundred miles while the bladder was traversing one; but, my dear Simplicio, this phenomenon which you adduce against my proposition is precisely the one which confirms it. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [117-118]: [117-118])

Galileo passes thus from "highly probable" following his limiting case argument, to "confirms" now. Here is how he argues with a constant cause, constant effect argument12 : SALV. [...] Let me once more explain that the variation of speed observed in bodies of different specific gravities is not caused by the difference of specific gravity but depends upon external circumstances and, in particular, upon the resistance of the medium, so that if this is removed all bodies would fall with the same velocity; and this result I deduce mainly from the fact which you have just admitted and which is very true, namely, that, in the case of bodies which differ widely in [specific] weight, their velocities differ more and more as the spaces traversed increase, something which would not occur if the effect depended upon differences of specific gravity. For since these specific gravities remain constant, the ratio between the distances traversed ought to remain constant whereas the fact is that this ratio keeps on increasing as the motion continues [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [118]: [118]) That is, when observing two bodies of different specific weights free-falling from a small and a high altitude, we realize that from a small altitude the difference in speed is so small that is barely observable, while from a high altitude the difference in their speed increases as the spaces traversed increase. Since from a constant cause we should get a constant effect, differences in specific weights, which remains constant, should cause the same variation of speed from small and high altitudes. Having observed that this variation of speed is not constant, but increases during the fall, we could conclude that differences in specific weights cannot cause this variation of speed, which should be caused by external factors; i.e. the resistance of the medium. Thus in void, when the medium's resistance is removed, all bodies would fall with the same speed.

But Simplicio remains unpersuaded that this difference in speed should be caused by the medium's resistance, in such a way that if removed, all bodies would fall at the same speed. He will thus question the reason as to why the same medium produces different effects with the increase of the altitude of fall. Since the medium does not change as well, it should also produce a constant effect: SIMP. Very well: but, following your own line of argument, if differences of weight in bodies of different specific gravities cannot produce a change in the ratio of their speeds, on the ground that their specific gravities do not change, how is it possible for the medium, which also we suppose to remain constant, to bring about any change in the ratio of these velocities? [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [118]: [118]) Which provides Galileo the opportunity to meet this "clever" objection by explaining how the effect of the medium's resistance increases with acceleration:

SALV. […]

There is […] an increase in the resistance of the medium, not on account of any change in its essential properties, but on account of the change in rapidity with which it must yield and give way laterally to the passage of the falling body which is being constantly accelerated. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [119]: [119]) That is, the medium's resistance is treated differently in the De Motu and the Discorsi: In the latter, the medium not only makes the body lighter as in the De Motu, it also has a frictional effect, which keeps on increasing until the falling body reaches its terminal velocity: "the speed [of the falling body] reaches such a point and the resistance of the medium becomes so great that, balancing each other, they prevent any further acceleration and reduce the motion of the body to one which is uniform and which will thereafter maintain a constant value." [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [118]: [118]).

The effect of specific weight is also treated differently in the De Motu and the Discorsi, in plenum and in void. In the former, the young Galileo defended that speed of fall is proportional to the specific weight difference between the mobile and the medium, which brought him to conclude that in void, where the medium's specific weight is null, speed is proportional to the mobile's specific weight. While in the Discorsi, the difference in speed that we observe in plenum for two free-falling bodies, of different specific weights, does not translate to a difference in speed in void, since:

[Specific] weight is the means employed by the falling body to open a path for itself and to push aside the parts of the medium, something which does not happen in a vacuum where, therefore, no difference [in speed] is to be expected from a difference of specific gravity. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [118]: [118]) Put differently, while the De Motu's theory could be written as follows V ~ Wb -Wm (see 3.1), the Discorsi's theory could be written as follows: V = Vo [Wb -Wm]/Wb (Wb and Wm are as before the specific weights of the body and the medium, Vo is the speed in void) 13 . Thus in void where Wm = 0, all bodies fall at the same speed Vo.

Finally, Galileo will provide an empirical test: since measuring this variation in speed of two bodies falling from small heights was technically impossible at his time, and so "if there be a difference it will be inappreciable", Galileo will propose to substitute these observations by observations on pendulums with equal bobs made from different material. In these experiments Galileo could "repeat many times the fall through a small height" in such a way that they become "not only observable, but easily observable" [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [128]: [128]). But this, as Palmieri notes, is a different story.

Ignoring the medium's resistance without assuming vacuum

We could now answer the question asked in the introduction, why is the TE restricted to bodies of the same material, as follows: Galileo, in restricting his TE's scenario to bodies of the same material, was able to isolate and eliminate absolute weight as a causal factor and to postpone his analysis of specific weight and the medium's resistance. That is, Galileo in his TE only addressed principle (i), without making any reference to the effects of the medium's resistance, which is described in Aristotle's principle (ii). In this section I aim to analyse if Galileo was justified in ignoring the medium's resistance, without assuming vacuum. In fact, this assumption which is usually legitimate if the context were different, could not be explicitly made by Galileo: the TE appears in a larger discussion concerning the existence of vacuum and the possibility of motion in void, which makes any explicit assumption of vacuum inadmissible in the TE 14 . [START_REF] Atkinson | Galileo and prior philosophy[END_REF] set out to analyse the speed (acceleration and terminal velocity) of fall of bodies in different situations: bodies of different material falling in plenum and in void, from the same altitude and different altitudes, from small and high altitudes, etc. This analysis, even it is irrelevant to and consistent with Galileo's TE as analysed here (since I defend that Galileo's TE is only refutational 15 ), remains interesting in its own right: it shows the complexity of taking into account all relevant causal factors -known (e.g. medium's resistance in plenum) or even unknown (e.g. inhomogeneous gravitational field of the earth, even in void) by Galileothat could affect the speed of free-falling bodies.

Some interesting parts of this analysis, which are directly related to Norton's quote in the introduction concerning Galileo's "hidden" assumption 8ai.e. speed of fall depends only on the body's weight -, are in fact explicitly addressed by Galileo in the Discorsi. This is first reflected in Galileo's choice of particulars involved in his TE's scenariobodies made of the same material and have the same shape differing only in size -and second by Galileo's analysis of a small effect of the medium's resistance affecting even these particulars.

First, directly after the TE, Galileo underlines that:

Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds which are proportional to their weights; this he illustrates by use of bodies in which it is possible to perceive the pure and unadulterated effect of gravity [i.e. absolute weight], eliminating other considerations […] which are greatly dependent upon the medium which modifies the single effect of gravity alone [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [109]: [109]) [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [109]: [109])

Put differently, Galileo is making Norton's hidden assumption 8a but without assuming vacuum. By his choice of particulars Galileo is considering a situation, like Aristotle did, in which absolute weight is the only causal factor and it is possible to perceive its pure and unadulterated effect. But Galileo is not making this assumption to argue from his TE that all bodies fall at the same speed, as in Norton's TE-argument (Norton 1996: 341-343)since even assuming 8a, bodies could fall proportionally to their specific weights as we have seen in (3.1) -, but to show that absolute weight could not be a causal factor, contrary to Aristotle's principle (i) or any other theory linking differences in speed to differences in absolute weight.

Second and more subtly, Galileo knew that even for these bodies, most, not all of the effects of the medium's resistance could be taken into account in his TE. Most, not all since one small effect of the medium's resistance remains disproportional for larger and smaller bodies. Indeed, just before the above quote, Galileo makes reference to this small effect when he claims that "[y]ou find, on making the experiment, that the larger outstrips the smaller by two finger-14 Indeed, assuming vacuum at this point in the TE could invite the Aristotelian to reject the TE on the ground that void is impossible. cf. El Skaf (2016) ch.7 for an analysis of how TEs could fail and El Skaf (2017) for an analysis of the notion of possibility at play in TEs. 15 Which seems in line with Atkinson and Peijnenburg analysis as summarized in 2007: "As a destructive thought experiment, refuting the Aristotelian theory of falling bodies, we deem Galileo's thought experiment to be unparalleled, one of the best. But as a constructive thought experiment, claiming that all bodies fall at the same rate, it has a serious flaw. For it fails to make explicit a hidden assumption that is not always applicable, namely that the rate of fall of a body depends only on its weight, and on nothing else." (207) breadths" and then dismisses this difference on the account that Simplicio would "not hide behind these two fingers the ninety-nine cubits of Aristotle".

Finally and most importantly, at the end of his argumentative strategy Galileo comes back to this small effect and sets out "to explain how one and the same medium produces such different retardations in bodies which are made of the same material and have the same shape, but differ only in size" [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [132]: [132]). For Galileo this explanation "requires a discussion more clever than" the previous explanations of the different effects of the medium's resistance. Galileo's solution lies in:

[T]he roughness and porosity which are generally and almost necessarily found in the surfaces of solid bodies. […] in the motion of falling bodies these rugosities strike the surrounding fluid and retard the speed; and this they do so much the more in proportion as the surface is larger, which is the case of small bodies as compared with greater. [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF](Galileo /1914: [132]: [132]) That is, the medium affects disproportionally even the two falling mobiles involved in the TE's scenario. The medium's resistance is more important the bigger the mobile's surface to absolute weight ratio is. The medium thus affects less the speed of fall of the larger mobile than that of the smaller one, since the former have a smaller surface to absolute weight ratio than the latter (for which Galileo provides a geometrical proof, [START_REF] Galileo | Dialogues Concerning Two New Sciences by Galileo Galilei[END_REF]Galileo /1914: [133-134]: [133-134]). The larger mobile will be less retarded by the medium and thus will have a greater speed. Which means that, in the context of the TE where Galileo is in no position to assume vacuum, the larger mobile falls faster than the smaller one. If we take this effect into account in the TE's scenario, then it is hard to see how even the destructive conclusion could be obtained. Galileo thus needed to ignore this small effect of the medium's resistance in order to refute Aristotle's principle (i).

If we don't ignore this effect of the medium's resistance, then how should we analyse the TE. One option, which rather complicates things, is to analyse how the two bodies are tied together: are they merely united or smelted together. This point has already been raised by [START_REF] Gendler | Galileo and the Indispensability of Scientific Thought Experiment[END_REF] and [START_REF] Atkinson | Galileo and prior philosophy[END_REF]. In analysing Norton's hidden assumption 8a and Gendler's reconstructed argument, Atkinson and Peijnenburg correctly explain that if we take into account this small effect of the medium's resistance, then things get complicated, since "two lead spheres of different weights (and therefore with different volumes), will have different terminal velocities. If they are tied together side-by-side, the terminal velocity of the united system will lie between the terminal velocities of the constituents [... and Galileo is justified in refuting Aristotle's principle (i)]. If, on the other hand, the spheres are melted and recast as one sphere of weight equal to the sum of the weights of the two original spheres, then the terminal velocity of the united system will be greater than those of either of the constituents. The reason [as we saw Galileo was aware] is that the retarding viscous force is a function of both the velocity and of the surface area of the falling body. The smelted sphere falls more quickly than the united spheres because the surface of the former is smaller than the combined surfaces of the latter." (p. 123) In this latter case, Galileo is no longer able to refute Aristotle's principle (i), since the smelted body falls faster than its constituents. That is, the mediativity principle no longer applies for smelted bodies.

I submit that there is no need to complicate the TE's analysis: Galileo is in a position to ignore this small effect of the medium's resistance since the TE appears in a larger argumentative strategy in which Galileo comes back to this effect and explains it.

Conclusion

Let us summarize and conclude. In this paper I aimed at clarifying the reasons behind Galileo's restriction, in the TE's scenario and conclusion, to bodies made of the same material. This restriction turned out to be of central importance to understanding the function and limit of the TE. I retraced the history of this TE to its first occurrence in the De Motu and showed that the TE is only refutational; it aimed at refuting Aristotle's principle (i) by showing that absolute weight could not be a causal factor.

I then exposed how Galileo, following the same TE, defended two incompatible theories of free fall and I argued that both theories could be traced to Galileo's analysis of specific weight: following a hasty Archimedean analogy, the young Galileo maintained specific weight as a causal factor and defended an early theory of free-fall according to which speed is proportional to specific weight, even in void. Five decades later and with two new arguments, Galileo eliminated specific weight as a causal factor and defended his final theory of free-fall according to which in void, all bodies fall at the same speed.

I finally showed that Galileo, in the TE, needed to ignore one small effect of the medium's resistance affecting even the kind of particulars involved in the TE's scenario: bodies made of the same material and having the same shape, differing only in size. This effect either complicates the TE if it is taken into account, or Galileo is justified in ignoring it only when we analyse the TE as part of a general argumentative strategy in which Galileo comes back to this small effect and explains it.

In conclusion, we could draw out from this historical analysis at least the following two implications for the debate on TEs.

First contra Norton's "irrelevant particulars" and elimination thesis (see ft 1), in appraising TEs it seems crucial to analyse the function(s) of some of the details involved in their scenarios, instead of trying to eliminate them. Indeed, some particulars involved in a TE's scenario have a crucial function. In Galileo's TE, they permit to isolate the effect of absolute weight without assuming vacuum. Of course some particulars are irrelevant, for instance the two falling bodies could be yellow or blue, weight 8 and 4 kg or 12 and 6 kg, however they should be made of the same material and have the same shape, if not absolute weight could no longer be isolated as a causal factor. In addition, following the TE and Norton's quote in the introduction, the equality in speed only applies to "special cases", that is to the kinds of particulars involved in the TE's scenario for which we could ignore the effect of the medium's resistance. Which is not the case for any two free-falling bodies, even of the same material. For instance, for bodies having a different shape, such as a nugget of gold and a leaf of gold (example given by Galileo), the medium's resistance could no longer be ignored without assuming vacuum.

Second contra Brown, it is clear that Galileo's TE could not, and even did not reveal, and a fortiori justify a law of nature -i.e. in void, all bodies fall at the same speed -platonically or otherwise. For the simple reason that following the TE, and even if we assume vacuum, we still don't know if bodies fall proportionally to their specific weight (1590) or not (1638). The TE, restricted to bodies of the same material having the same shape, allows Galileo to isolate and eliminate absolute weight as a causal factor, but remains silent concerning other causal factors, in particular specific weight (see 3). At most, the TE could lead to a weaker reading of the law; i.e. all bodies of the same material fall at the same speed. However, this conclusion follows if the medium's resistance is the only remaining causal factor (see answer to the last question below) and if we are justified in idealizing all of its effects, in such a way that we could say, following the TE, that a nugget of gold falls at the same speed than a leaf of gold, which amounts to assuming vacuum.

Finally, answers to the questions formulated in the introduction are found explicitly throughout the paper, let me restate them briefly here for clarity: What is the TE's function (or intended conclusion) for Galileo? To show that absolute weight is not a causal factor, thus refuting Aristotle's principle (i). What is its role in Galileo (1590 and 1638)'s argumentative strategies? To eliminate absolute weight as a causal factor, thus paving the way to analyse specific weight and the medium's resistance. What is the function of the particulars involved in its scenario? To isolate absolute weight in order to eliminate it as a causal factor. What are the idealisations involved? Void is not explicitly assumed, but one small effect of the medium's resistance is ignored. Are these idealisations justified? Yes, if we analyse the TE in the general strategy where this small effect is subsequently explained. Since vacuum could not be explicitly assumed in the TE and thus its scenario takes place in plenum, then how did Galileo take into account the multiple effects of the medium's resistance? Most of them were taken into account by Galileo's choice of particulars, one small effect was ignored but then later explained. In case we assume vacuum (for modern readers), what constructive conclusion could the TE lead to? At most, the TE could lead to the following: in void, all bodies of the same material fall at the same speed. Is this conclusion justified? Yes, if there are no remaining causal factors. But as shown in Atkinson and Peijnenburg (2004: 124-125) and unbeknown to Galileo, different causal factors could affect speed, even in void. For instance, the earth inhomogeneous gravitational field affects disproportionally the acceleration of bodies of the same material when they are dropped from different heights.

Fig. 1 :

 1 Fig.1: Galileo's De Motu analysis of the different ways inequalities in speed could come about

Prior to Galileo, we find a similar TE in the work of Jean Baptiste Benedetti (1553) who imagines a scenario involving the fall of two equal bricks, by themselves and then attached (cf.Koyré 1960: 203).

This division will be again introduced in the Discorsi. For instance when Simplicio explains Aristotle's argument against motion in void, he claims that Aristotle: "first supposes bodies of different weights to move in the same medium; then supposes, one and the same body to move in different media. In the first case, he supposes bodies of different weight to move in one and the same medium with different speeds which stand to one another in the

Cf. Palmieri 2005: 226-227 for a similar analysis

Galileo will separate, in the Discorsi, from the idea that we could understand falling bodies by analogy to what happens in a balance, since bodies become weightless during their fall, a balance falling with a body cannot

Cf. El Skaf and Imbert (2013) for a defence of unfolding as a general task of science involved in several tools (computer simulations, real experiments and TEs). Cf. El Skaf Rawad (2016) ch.7 for an account of how TE reveal and resolve inconsistencies through a common structure which involves mentally unfolding TEs' scenarios.

Cf.[START_REF] Koyré | Le De Motu Gravium de Galilée. De l'expérience imaginaire et de son abus[END_REF] and[START_REF] Van Dyck | An Archeology of Galileo's science of motion[END_REF] for a similar formula.

This is also explicit inGalileo's Postils to Rocco (ca. 1634-1635) where he also uses the same TE and arguments as in the Discorsi (cf.Palmieri 2005: 232-233).

Cf.[START_REF] Koyré | Le De Motu Gravium de Galilée. De l'expérience imaginaire et de son abus[END_REF], p. 213 for a similar analysis

cf.[START_REF] Koyré | Le De Motu Gravium de Galilée. De l'expérience imaginaire et de son abus[END_REF] and[START_REF] Van Dyck | An Archeology of Galileo's science of motion[END_REF] for a similar formula.