The chapter begins with the following statement: A body can be conceived as extended in length, width and depth. Thus, every body has three dimensions. How do one get this idea? By abstraction, is the simple answer. Our mind (“esprit”) has the power to made abstractions («mais notre esprit ayant le pouvoir de faire des abstractions»). This ability for abstraction allows us to imagine one-dimensional or two-dimensional figures, too. To mention a simple case: Given two points A and B at a certain distance from each other; a distance is defined as an interval that separates each points. Two points generate, or form a line. How can we imagine this? According to Du Châtelet, point A travelling to meet point B leaves a production of itself in each part of the interval.

Imaginations are based on abstractions. They should not be confused with physical bodies. If one ignores the difference between the abstractions of our mind and physical bodies, one confuses the divisibility of a geometrical body with the divisibility of a physical body, and thus enters the labyrinth of the divisibility of the continuum (InstPhy, § 168):

La plûpart des Philosophes ayant confondu les abstractions de notre esprit avec le Corps Physique, ont voulu démontrer la divisibilité de la Matiére à l’infini, par les raisonnemens les es Géomètres sur la divisibilité des lignes qu’on pousse jusqu’à l’infini; mais ils se seroient épargné toutes les difficultés que cette divisibilité entraîne, s’ils avoient prissoin de ne jamais appliquer les raisonnemens que l’on fait sur la divisibilité du Corps Géometrique, aux Corps naturels & Physiques. (Amsterdam 1742)

Most philosophers, having confused the abstraction of our mind with physical body, have wanted to demonstrate the divisibility of matter to infinity by means of the reasoning of the geometers on the divisibility of lines that one pushes to infinity. This has given rise to the famous labyrinth of the divisibility of the continuum that has so embarrassed the philosophers. But they could have avoided all the difficulties that this divisibility involves if they had taken care never to apply the reasoning that one applies to the divisibility of the geometrical body to natural and physical bodies. (Copyright © 2014 LP)

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

The geometrical body has no determinate and actual parts. It is infinitely divisible, because the number of its possible parts are indeterminate and can be arbitrarily choosen. According to Du Châtelet, arguments against the possibility of motion, like Zenon’s paradox, confuse geometrical extension with physical extension (InstPhy, § 171):

Le plus fameux de tous ces Sophismes étoit celui que Zenon avoit appellé l’Achille, pour marquer sa force invincible. Il supposoit Achille courant après une Tortue, & allant dix fois plus vite qu’elle, il donnoit une lieue d’avance sur lui, celle ci parcourera un dixième de lieue; pendant qu’il parcourera ce dixième, la Tortue parcourera la centième partie d’une lieue; ainsi, de dixième en dixième, la Tortue devancera toujours Achille, qui ne l’atteindra jamais. (Amsterdam 1742)

The most famous of all the paradoxes was that of Zeno had called the Achilles, in recognition of Achilles’ invincible force. He proposed Achilles running toward a tortoise, and, as Achilles ran ten times faster than the tortoise, he gave the tortoise a head start of one league, and he reasoned as follows: while Achilles traveled one tenth of a league, the tortoise would travel one hundredth of the league; thus, from tenth to tenth, the tortoise would always remain in front of Achilles, who would never catch up with it. (Copyright © 2014 LP)

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

First, it does not follow from this thought experiment that motion is impossible, but at best that Achilles never catches up with the tortoise. But that is also wrong. Because the paradox is based on a false assumption as Grégoire de Saint-Vincent has shown (InstPhys, § 171):

Grégoire de Saint Vincent fut le prémier qui en démontra la fausseté, & qui assigna le point précis, auquel Achille doit atteindre la Tortue, & ce point se trouve par le moyen des progressions Géométriques infinies, au bout d’une lieue & d’unneuvième de lieue: car la somme de toute progression Géométrique infinie décroislante est finie; & cela, parce qu’être infini, ou s’étendre à l’infini, sont deux choses très différentes. (Amsterdam 1742)

Grégoire de Saint-Vincent was the first to demonstrate its falsity, and who assigned the precise point at which Achilles would have to reach the tortoise, and this point was found, by means of infinite geometrical progressions, to be one and one ninth leagues. The sum of any infinite geometrical progression is finite, and this is because the infinite being, and infinite extension, are two very different things. (Copyright © 2014 LP)

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

In his Opus geometricum quadraturae circuli sectionum coni (1647), Grégoire de Saint Vincent resolved Zeno’s Achilles paradox, by summing an infinite geometric series. The flaw in Zeno’s argument is his unstated assumption that the sum of an infinite cannot be finite. Du Châtelet’s reasoning for this fact goes beyond Grégoire de Saint Vincent’s proof. She states that an infinite being and infinite extension are two different things. Further, the physical divisibility of an extended body ad infinitum is not possible. By contrast, there is nothing wrong with the geometrical divisibility of extension (InstPhy, § 171):

Ainsi, la divisibilité de l’étendue à l’infini est en même tems une vérité Géométrique, & une erreur physique. (Amsterdam 1742)

Thus, the divisibility of extension to infinity is at the same time a geometrical truth and a physical error. (Copyright © 2014 LP)

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

This argument is directed against John Keill. In Introductio ad veram physicam (1701) Keill had claimed that the infinite divisibility of matter can be demonstrated by geometry, and that one could fill the entire universe with a grain of sand. Du Chatelet disagrees. According to her opinion one must not admit anything into physics besides actual parts, whose existence can be demonstrated by experience or by rigorous demonstration.

In the following passages a number of examples are described to support the correctness of this thesis: experiments with the microscope, experiments with electricity, the work of the drawer and beaters of gold. It is noteworthy that paragraphs 172-183 of the Paris edition 1740 and of the Amsterdam edition 1742 do not correspond with each other. For example, in § 177 of the Paris edition 1740 two kinds of matter (proper matter and foreign matter) are distinguished. Proper matter is defined as constant, foreign matter (like air, water etc.) as variable. One can find the same classification in Christian Wolff’s Vernünfftige Gedancken von den Würckungen der Natur (1723, § 14 ff.). However, this distinction is not made in the Amsterdam edition 1742. Instead, a distinction is made between primitive and derivative matter. This classification is missing in the Paris edition 1740. A detailed comparison between the editions including the manuscript would go beyond the scope of this Reader. As a remarkable example, the last paragraph on the ninth chapter of the Paris edition (following the manuscript) and Amsterdam edition is quoted here:

Mais comme il nous reste peu d’espérance de découvrir les Matiéres plus simples par les mixtions desquelles les Corps sensibles résultent, un Physicien qui ne veut pas perdre son tems, doit se contenter de découvrir les raisons les plus prochaines, que l’industrie humaine peut appercevoir, & n’admettre de Matiéres & de Mouvemens, que ceux dont l’éxistence peut être démontrée. (Paris 1740, § 185).

But as we are left with little hope of discovering the simple matters, the mixture of which results in sensible bodies, a physicist who does not wish to waste his time must content himself with discovering the closest reasons that human industry can perceive, and will admit only matters and motions the existence of which can be demonstrated. (Copyright © 2014 LP)

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

Vous pouvez conclure de tout ce qui a été dit dans ce Chapitre, que bien qu’il soit très important en Métaphysique de savoir qu’il ne peut y avoir d’atomes physiques, & que toute étendue est à la fin composée d’êtres simples, cependant ces questions n’ont qu’une influence très éloignée dans la Physique expérimentale, ainsi le Physicien peut faire abstraction des différens sentimens des Philosophes sur les élémens de la matière, sans qu’il en résulte aucune erreur dans ses expériences, & dans ses explications, car nous ne parviendrons jamais ni aux Etres simples, ni aux atomes. (Amsterdam 1742, § 183)

From everything I have said in this chapter, you can conclude that although it is very important in metaphysics to know that there can be no physical atoms, and that all extended things finally consist of simple things, these questions nevertheless have a very small influence regarding the experiments of natural science. So the physicist can ignore the philosophers’ different opinions of the elements of matter without being erroneous in his experiments and explanations. Because we’ll never get to the simple things and atoms. (my own translation)

As different as the two versions are, both editions emphasize the same thing: We cannot know anything about simple beings. We don’t need to know anything about this either. In several passages of the chapter a number of examples and experiments are described to support this claim. In the Amsterdam edition of 1742, Du Châtelet refers, among others, to Nicolaas Hartsoeker’s hypothesis of the preexistence of germs (Essai de Dioptrique 1694) and to Robert Boyle’s experiments upon the dissolution of gold in his solvent (The Philosophical Works of the Honourable Robert Boyle, first published in London 1725). Despite the limitations on what we may know – we cannot go beyond the phenomenal which is subject of our senses and finite experience – knowledge, based on experience and solid reasoning, is still possible.

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See also the entry DIVISIBILITÉ, (Géom. & Phys.) in d’Alembert’s and Diderot’s Encyclopédie. See also the entry “Divisibilité” 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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