At the beginning of the fifth chapter, Du Châtelet stresses that the question of the nature of space is one of the most famous, controversial and essential question in physics and metaphysics. Some argue that space is nothing over and above things; space cannot be a real thing in-itself, it is a mental abstraction, an ideal being, the order of coexisting things; and there is no space without bodies (InstPhy, § 72):

Some have said: Space is nothing over and above things, it is a mental abstraction, an ideal Being, it is nothing other than the order of things as they coexist, and there is nothing to space except bodies [there is no space without bodies]. (Copyright © 2018 KB)

Quelques-uns ont dit: l’Espace n’est rien hors des choses, c’est une abstraction mentale, un Etre idéal, que ce n’est que l’ordre des choses entant qu’elles coexistent, & qu’il n’y a point d’Espace sans corps. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Others disagree. They maintain that space is absolute and real. Space has to be distinguished from the bodies placed in it. Space is impalpable extended, penetrable, and not solid, a universal vase, or vessel, within which bodies are placed (InstPhy, § 72):

Others have, on the contrary, maintained that Space is an absolute Being, real, and distinct from the bodies placed in it, i.e. an impalpable, penetrable, non-solid extension, the universal vessel that receives the Bodies that are placed in it; in a word, a kind of immaterial and infinitely extended fluid, in which the Bodies swim. (Copyright © 2018 KB)

D’autres au contraire ont soûtenu, que l’Espace est un Etre absolu, réel, & distinct des corps qui y sont placés; que c’est une étendue impalpable, pénetrable, non solide, le vase universel qui reçoit les Corps qu’on y place; en un mot, une espèce de fluide immatériel & étendu à l’infini, dans lequel les Corps nagent. (Amsterdam 1742)

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The idea that space is distinct from matter dates back to the Pre-Socratic philosophers Leucippus and Democritos. Pierre Gassendi revived the idea of space as an immaterial and incorporeal being, impalpable and incapable of action und passion. John Locke also distinguished pure space from the bodies that fill it. In his Essay Concerning Human Understanding (1690; French translation 1700, revised edition 1720) he defined impenetrability as the criterion for distinguishing space and bodies: bodies are impenetrable, space is penetrable. Another difference between space and bodies is that we cannot see and touch space, but we can see and touch bodies. John Keill held the same opinion. In his book Introductiones ad veram Physicam (first published in Latin 1701,  translated into English under the title An Introduction to Natural Philosophy 1720) he goes even further. He claims to be able to prove that a larger amount of empty space actually exists, even within matter itself (Keill 1720, 117):

Let us suppose now two Globes of equal Magnitudes, the one of Lead, the other of Cork; if the Quantity of Matter in both was the same, (by what has be shewn [sic]) both Bodies would equally ponderate: for the subtilest Matter occupying the Pores of the Cork, would penetrate equally with the Matter of Lead that is equal to it. But since there is a great difference in the Weights of these two Bodies, there will be also a great difference in the Quantity of their Matter; and if Lead is thrice heavier than Cork, the Matter contained in the Lead will be triple of that in the Cork: so there will be more Pores and Spaces absolutely empty in the Cork,  than in the Lead. A Vacuum therefore is not only possible, but actually given.

Christian Wolff rejected Keill’s argument. He maintained that the so-called “materia subtilissima” did not increase the weight of the body. According to the Archimedean principle, the bodies would lose as much weight as the weight of the volume of the fluid they displaced (Aerometriae Elementa 1709). Keill countered in a letter to Wolff printed in January 1710 (Acta eruditorum Jan. 1710, pp. 11-15). Wolff answered promptly. His reply appeared in the same journal one month later (Acta eruditorum Feb. 1710, 78-80). Johann Bernoulli I intervened. In a letter to Wolff  (26 April 1710), he agreed with Wolff that Keill’s argumentation was wrong. However, Bernoulli I argued differently from Wolff: If the greater weight of the lead ball would be due to the fact that it loses less weight in the “fluidum subtilissimum” than the cork ball of the same size, then the resulting weight difference would be so small that it would not be observable. Lead and cork thus have almost the same specific weight, which contradicts experience as well as Keill’s argumentation. The latter was based on the assumption that the weight of the body is proportional to the quantity of matter.

Du Châtelet was familiar with this controversy which led to the question: How does matter relate to gravity in an empty space? Du Châtelet argues that empty space cannot exist. Because atoms, or first particles of matter, have sizes and shapes. Empty space cannot explain why the particles have this figure rather than another possible one, and why they are of a certain size (InstPhy, § 73). Consequently, physical space is incompatible with the hypothesis of an absolute empty space. Despite her rejection of empty space, Du Châtelet did not abandon Newton’s theory of gravity. In the following paragraphs she focuses on some key aspects of Newton’s gravitational theory and concept of space (InstPhy, § 73):

Several mathematicians have embraced the opinion of absolute void on the authority of Mr. Newton. This great man believed, according to Mr. Locke, that one can explain the creation of matter through Space, imagining that God would have rendered several regions of Space impenetrable: one sees in the General Scholium, which is at the end of Mister Newton’s principles, that he believed that Space was the immensity of God, in the Opticks he calls it the God’s Sensorium; that is to say, it is that through which God is present in all things. (Copyright © 2018 KB)

L’autorité de M. Newton a fait embrasser l’opinion du vuide absolu à plusieurs mathématiciens. Ce grand homme croyoit, au rapport de M. Loke, qu’on pouvoit expliquer la création de la matière, en supposant que Dieu auroit rendu plusieurs parties de l’espace impénétrables: on voit dans le scholium generale, qui est à la fin des principes de M. Newton, qu’il croyoit que l’espace étoit l’immensité de Dieu; il l’appelle, dans son optique le sensorium de Dieu, c’est-à-dire, ce par le moyen de quoi Dieu est présent à toutes choses.

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

In his General Scholium, added to the Principia in 1713, Newton had asserted that God, the Creator and the “lord of all” (“Pantocrator”) constitutes duration and space. The infinite dominion of the One God (in His omnipresence) is everywhere. In the “Queries” appended to his Opticks, Newton had claimed that space is the sensorium of God.  Locke even went further and claimed that Newton had explained the creation of matter through Space by God’s nature (see Pierre Coste’s translation Essai philosophique concernant I’Entendement humain 1700). Newton’s views on space and God gained popularity through the exchange of letters between Leibniz and Clarke. Du Châtelet was well aware of the Leibniz-Clarke correspondence (InstPhys, § 74):

Mr. Clarke has taken a great deal of trouble to support the opinions of Mr. Newton, as well as his own views on absolute Space, against Mr. Leibniz, who maintained that Space was nothing but the order of coexisting things. (Copyright © 2018 KB)

Mr. Clarke s’est donné beaucoup de peine pour soutenir le sentiment de M. Newton, & le sien propre sur l’espace absolu, contre M. Leibnitz qui prétendoit que l’espace n’était que l’ordre des choses coéxistantes. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Du Châtelet argues, in line with Leibniz, that the principle of sufficient reason banishes absolute space from the universe (InstPhy, § 74):

if Space is a real Being and subsistent without Bodies that could be placed in it, it makes no difference in which part of this homogeneous Space one places them, as long as they keep the same order among them: therefore there would not have been any sufficient reason why God would have placed the Universe in the location where it is now, rather than in any other, since he could have placed it 10,000 leagues further away, and put the East where the West is; or indeed he could have reversed it, so long as he kept things in the same place in relation to each other. (Copyright © 2018 KB)

car non seulement il n’y auroit, comme vous venez de le voir, aucune raison de la limitation de l’étendue; mais, si l’Espace est un Etre réel & subsisiant sans les Corps, & qu’on puisse les y placer; il est indifférent dans quel endroit de cet Espace similaire on les place, pourvu qu’ils conservent le même ordre entre eux; ainsi il n’y auroit point eu de raison suffisante pourquoi Dieu auroit placé l’Univers dans la place où il est maintenant, plutôt que dans toute autre , puisqu’il pouvoir le placer dix mille lieues plus loin, & mettre l’Orient où est l’Occident; ou bien enfin le renverser, en faisant garder aux choses la même situation entre ells. (Amsterdam 1742)

Leibniz used this thought experiment at the beginning of his third letter to Clarke. Leibniz argued that “tis impossible there should be a reason, why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner, and not otherwise; why everything was not placed the quite contrary way, for instance, by changing East into West” (Third letter to Clarke, § 5 [Feb 25, 1716]; L 682):

quil est impossible qu’il y ait une raison pourquoi Dieu, gardant les mêmes situations des corps entre eux, ait placé les corps dans l’Espace ainsi et non pas autrement et pourquoi tout n’a pas été pris à rebours (par exemple), par un échange de l’orient et de l’occident. (Des Maizeaux 1720, § 5, p. 32)

Du Châtelet reproduces Leibniz’ argument, which is a reductio ad absurdum argument, by constructing a situation in which God has to make a choice lacking a sufficient reason. She uses the principle of sufficient reason in order to argue against the deification of space and emphasizes that God’s will in itself constitutes sufficient reason for any act. Contrary to Clarke, who argued that God is entirely free to do as he pleases, Leibniz emphasized that everything God does must be done for a reason. If there is no sufficient reason for God’s acting in a particular manner, then he is no more than an all-powerful tyrant inventing arbitrary standards and acting in arbitrary ways. If there were indiscernibles, God would have violated the principle of sufficient reason, by choosing without reason between equally good alternatives. So, it would be absurd for God to act against reason because this would mean to act against his own nature. Against this background, Du Châtelet comes to the following conclusion (InstPhy, § 74):

Mr. Leibniz’s reasoning against absolute Space is therefore irrefutable, and one is forced to abandon this Space, if one does not wish to renounce the principle of sufficient reason; that is to say, to renounce the foundation of all truth. (Copyright © 2018 KB)

Ainsi le raisonnement de Mr. de Leibnits contre l’Espace absolu est sans replique, & l’on est forcé d’abandonner cet Espace, ou de renoncer au principe de la raisson suffisante, c’est-à-dire, au fondement de toute vérite. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

After presenting Leibniz’s argument against absolute space, Du Châtelet discusses three objections against the plenum and rejects these arguments. Du Châtelet herself favors a dynamic ether model in order to avoid the hypothesis of a force which acts through empty space. This ether model is based on a concept of matter, which is incompatible with the assumption that matter is dead and motionless.

The following paragraphs deal with the question how we came to form our ideas of extension, space, and continuity. Du Châtelet’s states that everything that we consider to be different from us we see as external to us, even though we know well that the idea we have of it exists within us. Consequently, “extension” can be defined as follows (InstPhy, § 77):

It follows from this that we cannot represent to ourselves several different things as being one, without this resulting in a notion that is attached to this diversity and union, and this notion we call Extension. (Copyright © 2018 KB)

l suit de-là que nous ne pouvons point nour représenter plusieurs choses différentes comme faisant un, sans qu’il en résulte une notion attachée à cette diversité & a cette union des choses, & cette notion nous la nommons Etendue. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

For example, we give extension to a line, insofar as we pay attention to several distinct parts which we see as existing externally to one another, which are united together and which are for this reason a single whole. One can find the same example in Christian Wolff’s Ontologia: “Linea rectae tribuitur extension, quatenus in ea distinguimus partes plures diversas atque adeo extra se invicem existentes, quae inter se unitae unum quid efficient” (Wolff 1730, § 548). In this sense, the concept of extension corresponds to that of a surface, or geometric body. The line, for example, may be divided into as many parts as one wishes, it will be always the same line. The same holds for surfaces and for geometrical bodies (InstPhy, § 78):

So all of extension must be conceived of as uniform, homogeneous, and having no internal determination that distinguishes one part from another, since if we place these parts however we wish, the result will always be the same Being, and that is how we arrive at the idea of absolute Space, which we consider to be homogeneous and indiscernible. (Copyright © 2018 KB)

Ainsi toute l’étendue doit être conçue comme étant uniforme, similaire, & n’ayant point de détermination interne qui en distingue les parties les unes des autres, puisque étant posées comme l’on voudra, il en résultera toujours le même être; & c’est de là que nous vient l’idée de l’espace absolu que l’on regarde comme similaire & indiscernable. (Amsterdam 1742)

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If one starts from the question of how we form the concept of extension and gain the concept of space from it through abstraction, one inevitably led to represent space as a container, i.e., a substance independent of the beings that are placed in it (InstPhy, § 80):

insofar as it is possible by an act of our understanding (acte de l’entendement) , to restore to these Beings the determinations that we have stripped from them by abstraction, it seems to the imagination that we are importing something that had not been there before; and so this Being appears to be modifiable. (Copyright © 2018 KB)

en tant qu’il est possible, par un acte de l’entendement, de rendre à ces êtres les déterminations dont nous les avons dépouillées par abstraction, il semble à l’imagination que nous y transportons quelque chose qui n’y étoit pas, & alors cet être nous paroît modifiable. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

The concept of extension as described above fits perfectly the classical definition of geometry as a science of continuous magnitudes and figures, a science of triangles and squares, circles and conic sections, parallel lines and obtuse angles, i.e., the three-dimensional, homogeneous, and infinite Euclidean space. In Leibniz’ theory, however, space did not anymore constitute an independent background, but takes the form of a geometrical object itself. Leibniz’s aim with his analysis situs was to represent the relations among geometrical figures directly, without recourse to the Cartesian co-ordinates and equations of ordinary analysis. The basis for his new approach to space was to define an extensum (extended thing) in terms of “situs,” i.e., situation. Initially, “situs” denoted a disposition of smallest parts or unities in relation to the whole. The basic relations are those of equality, similarity and congruence. Congruence presupposes co-existence. This leads Leibniz to a new definition of extensum as a whole with co-existing parts that have mutual situation and to a more sophisticated characterization of space in terms of extensum and situs: it is the ordering of situations that constitutes space. As Leibniz said in his correspondence with Samuel Clarke:

I have said more than once that I hold space to be something merely relative, as time is: that is, I hold it to be an order of co-existences as time is an order of successions. For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together, without inquiring into their particular manner of existing. (Third letter to Clarke, § 4 [Feb 25, 1716]; L 682)

J’ay marqué plus d’une fois, que je tenois l’Espace pour quelque chose de purement relatif, comme le Temps; pour un ordre des Coexistences, comme le Temps est un ordre de Successions. Car l’espace marque en termes de possibilité, un Ordre des choses qui existent en même temps, en tant qu’elles existent ensemble; sans entrer dans leurs manieres d’exister. (Des Maizeaux 1720, § 4, p. 31)

The historical background for Du Châtelet’s considerations was undoubtedly the correspondence between Leibniz and Clarke. Leibniz had asked Clarke “how men come to form to themselves the notion of space.” We observe, to paraphrase Leibniz, a series of coexisting things. We abstract from these observations and thus gain the concept of space. Du Châtelet was obviously familiar with Leibniz’s idea of an analysis situs, which should be capable of finding new truths and new relations. She refered to Leibniz’s concept of fictions as “one of the greatest secrets of the art of invention, and one of the greatest resources for the solution of the most difficult problems” (InstPhy, § 86). From a Leibnizian perspective, “we are right to define space”  (InstPhy, § 79):

the order of Coexisting things, that is to say, the similarity in the manner in which Beings coexist. For the idea of Space arises from our attending solely to their manner of existing externally to one another, and representing to ourselves that this coexistence of several Beings produces a certain order or similarity in their manner of existing; and so once one of these Beings is taken to be the first, another becomes the second, another the third, and so on. (Copyright © 2018 KB)

Ainsi, l’on a raison de définir l’Espace l’ordre des Coexistans, c’est-à-dire, la ressemblance dans la manière de coexister des êtres; car l’idée de l’espace naît de ce que l’on ne fait uniquement attention qu’à leur manière d’exister l’un hors de l’autre, & que l’on se représente que cette coexistence de plusieurs êtres produit un certain ordre ou ressemblance dans leur manière d’exister; ensorte qu’un de ces êtres étant pris pour le premier, un autre devient le second, un autre le troisième, &c. (Amsterdam 1742)

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If one understands space as the order of coexisting things, belonging to the concept of extension, the principle of continuity has to be taken for granted. In § 81, Du Châtelet has defined continuity as follows:

We call a Being continuous when its parts are arranged one after another, such that it is impossible to place others in a different order between two of them, and in general we conceive of continuity whenever we cannot place anything between two parts. (Copyright © 2018 KB)

Nous appellons un Etre continu lors’qu’il a des parties rangées les unes auprès des autres, ensorte qu’il soit impossible d’en ranger d’autres entre deux dans un autre ordre. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

In his German Metaphysics (1720, § 59), Wolff defined continuity in a similar way:

Wenn die Theile dergestalt in ihrer Ordnung auf einander folgen, daß man zwischen ihnen nicht andere in einer andern Ordnung setzen kann; so saget man, es gehe in einem fort, und heißet ein auf solche Art zusammen gesetztes Ding ein stetiges Ding.

Wolff used the same example as Du Châtelet did in order to explain the content and meaning of this definition: the shine of a mirror is continuous, because we cannot see any unpolished parts between the parts of the glass that interrupt its continuity. Du Châtelet also distinguishes between “continuous” and “contiguous” (InstPhy, § 81):

But when two parts of extension simply touch and are not joined to each other, in such a way that there is no internal reason why we could not separate them or put something else between them, such as cohesion or pressure from surrounding Bodies, so we call them contiguous. (Copyright © 2018 KB)

mais lorsque deux parties d’étendue se touchent simplement & ne sont point liées ensemble, en sorte qu’il n’y a point de raison interne, comme celle de la cohésion, par exemple, qui empêche de les séparer, & de mettre quelqu’autre chose entre- deux , alors on dit que ces parties sont contigues. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

In the case of contiguity, parts are separated in actuality, in contrast to continuity, where the separation is no more than a possibility. Two hemispheres of lead, for example, are two actual parts of the sphere of which they are two halves, that is actually separated and divided into two parts, which will become contiguous if we place them one next to the other so that there is nothing in between them. But if we were to reunite them by fusion into a single whole, then this whole would become a continuity.

The distinction between contiguity and continuity dates back to Aristotle and was redefined by Leibniz who assumed that the phenomenal world is a dense contiguum and not a real continuum.  Compare Christian Wolff: “Contigua cur non sint continua” (Ontologia 1730, § 558).

Space appears to us as homogeneous, uniform, continuous, void and penetrable, immutable and eternal, infinite and limitless, in short, the universal vessel that contains all things. However, all these properties that we attribute to space have no reality but in the abstractions of our mind (“esprit”). Such abstractions might be helpful in the search for truths, but they become very dangerous if we take them for absolute real and mind-independent. Du Châtelet concludes (InstPhy, § 87):

Since the Abstract cannot subsist without a Concrete thing, that is to say without a real and determined Being from which we are abstracting, it is certain that there is Space only insofar as there are real and coexistent things. (Copyright © 2018 KB)

cependant, comme l’Abstrait ne peut subsister sans un Concret, c’est-à-dire, sans un Etre réel & déterminé duquel on fait l’abstraction, il est certain qu’il n’y a d’Espace qu’entant qu’il y a des choses réelles & coëxistantes. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Du Châtelet compares the concept of space with the concept of number, by using the following analogy (InstPhy, § 87):

Space is to real Beings as Numbers are to numbered things. (Copyright © 2018 KB)

l’Espace est aux Etres réels, comme les Nombres aux choses nombrées. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Space is an abstraction of coexisting things as numbers are an abstraction of things numbered. It would be meaningsless to speak about numbers without things that can be numered. Analogous, space would be meaningsless without postulating something that act in space (and time). The same analogy one can find in Christian Wolff’s Philosophia prima, sive Ontologia: “spatium se habet ad res simultanas sicuti numerus ad res numerates” (Wolff 1730, § 613).

Du Châtelet’s Institutions end with a distinction between “location,” “place” and “situation.” Location is not the placed thing itself; but it differs from the placed thing as an abstract thing from its concrete counterpart. The location of a being is its determined manner of coexisting with other beings (InstPhy, § 88). It can be absolute or relative. Absolute location is the one that suits a being insofar as we consider its manner of existing with the whole universe considered as immobile; and its relative location is its manner of coexisting with some particular beings. Imagine, for example, a person inside a ship which is sailing at constant speed. Since the ship is moving at constant speed and direction the preson will not feel the motion of the ship. The relative location on the boat does not change at all; but the person’s absolute location changes at each moment, because all the parts of the boat equally change their manner of existing with respect to the shore which we regard as immobile. However, if the person walks on the boat,  the relative and absolute location is changed at the same time.

We call place the assembly of several locations, that is to say all the locations of the parts of a body taken together (InstPhy, § 92). Finally, we call situation the order that several coexistent non-contiguous things maintain through their coexistence, such that taking one of them as the first, we give a situation to others that are far away in relation to that one. Thus, taking a house in a city as the first, all the others acquire a situation with respect to this house, because they are separated from each other, and because we can determine their situation by their distance from that which we took as the first. Therefore, two things have the same situation with respect to a third when they are at the same distance (InstPhy, § 93).

Motion, in this sense, means changing place or position. But that is not a sufficient explanation of how things change places.  Du Châtelet concludes (InstPhy, § 88):

Thus, in order to make certain that a Being has changed its place, and in order for this change to be real, the reason for its change, that is to say the force that produced it, must be in the Being at the moment at which it moves, and not in the coexisting Beings. This is because if we ignore where the true reason of change lies, we also ignore the reason why these Beings changed place. (Copyright © 2018 KB)

Ainsi, pour que l’on puisse assurer qu’un Etre a changé de lieu , & pour qu’il en change réellement, il faut que la raison de son changement, c’est-à-dire, la force qui l’a produit , soit en lui dans le moment qu’il se remue, & non dans les coéxistans; car si on ignore où est la véritable raison du changement , on ignore aufli lequel de ces Etres a changé de lieu.

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

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See also the entry ESPACE, subst. m. (Métaphys.) in d’Alembert’s and Diderot’s Encyclopédie. See also the entry “Espace” in: Johann Heinrich Samuel Formey (ed.): Dictionnaire instructif, où l’on trouve les principaux termes des sciences et des arts dont l’explication peut être utile ou agréable aux personnes qui n’ont pas fait des études approfondies. Halle: Gebauer 1767.

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