In her preface, Du Châtelet has compared hypotheses with the scaffolding of a building (InstPhy, XI). In the fourth chaper it becomes clear, why she uses this metaphor (InstPhy, § 53):

The true causes of natural effects and of the phenomena we observe are often so far from the principles on which we can rely and the experiments we can make that one is obliged to be content with probable reasons to explain them. Thus, probabilities are not to be rejected in the sciences, not only because they are often of great practical use, but also because they clear the path that leads to truth. (Copyright © 2009 BZ)

Les véritables causes des effets naturels & des Phénomènes que nous observons, sont souvent si éloignées des principes fur lesquels nous pouvons nous apuyer, & des Expériences que aire , qu’on est obligé de se contenter de raisons probables pour les expliquer: les probabilités ne sont donc point à rejetter dans les sciences, non seulement parce qu’elles sont souvent d’un grand usage dans la pratique, mais encore parce qu’elles frayent le chemin qui méne à la verité.

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

Scientific research begins with hypotheses. Why? We are mistaken and must be able to be mistaken in order to discover unknown truths and gain knowledge. For this reason, we should not banish hypotheses. This would hinder scientific progress.

Du Châtelet explicitly set up her position on hypotheses in opposition to both, the Cartesians and Newtonians. Both positions were too extreme and unjustified (InstPhy, § 55):

Descartes, who had established much of his philosophy on hypotheses, because it was almost impossible to do otherwise in his time,  gave the whole learned world a taste for hypotheses; and it was not long before one fell into a taste for fictions. […] Others have fallen to the opposite extreme. Disgusted with suppositions and errors that they found filled books of philosophy, they rose up against hypotheses and tried to make them suspect and ridiculous. (my own translation)

Descartes qui avoit établi une bonne partie de sa Philosophie sur des hypotheses parce qu’il étoit presqu’impossible de de faire autrement de son tems, mit tout le Mone sçavant dans le goût des hypotheses; & l’on ne sut pas longtems sans tomber dans celui des fictions. […] D’autres sont tombés dans l’excès contraire: degoutés des suppossitions, & des erreurs don’t ils trouvoient les livres de Philosophie rem plis, ils se sont élevés contre les hypothèses, ont tâché de les rendre suspectes & ridicules. (Amsterdam 1742)

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

In the first edition (Paris 1740) and in the manuscript, Newton is explicitly mentioned, who had asserted that he do not “feign” hypotheses. In the second edition (Amsterdam 1742) his name disappeares from the text and is replaced by a more vague reference: “D’autres sont tombés dans l’excès contraire.”

Against the Newtonians, Du Châtelet argued that our knowledge is not absolute certain, but probable. Against the Cartesians, she argued that probable knowledge has to be distinguished from fictions and imaginations. Thus, we need hypotheses because our knowledge lies between mere fictitious speculation and absolute certainty. Setting out from this premise, Du Châtelet defines the concept of hypothesis as follows (InstPhy, § 56):

When certain things are used to explain what has been observed, and though the truth of what has been supposed is impossible to demonstrate, one is making a hypothesis. Thus, philosophers frame hypotheses to explain the phenomena, whose cause cannot be discovered either by experiment or by demonstration. (Copyright © 2009 BZ)

Lorsque’on prend certaines choses pour rendre raison de ce qu’on observe, & que l’on n’est pas encore en état de démontrer la verité de ces choses que l’on a supposées, on fait une hypothese. Ainsi, les Philosophes établissent des hypotheses pour expliquer par leur moyen les Phénomenes dont nous ne sommes point en état de découvrir la cause par l’Expérience, ni par la démonstration. (Amsterdam 1742)

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

Hypotheses explain phenomena. To be more precise, a hypothesis is a proposed and provisional explanation for a phenomenon. Paragraph 60 supplements this definition (InstPhy, § 60):

So making hypotheses is allowed, and it is even very useful in all cases when we cannot discover the true reason for a phenomenon and the attendant circumstances, neither a priori, by means of truths that we already know; nor a posteriori, with the help of experiments. (Copyright © 2009 BZ)

Il est donc permis, & il est même très utile de faire des hypothèses dans tous les très utiles, cas, où nous ne pouvons point découvrir la veritable raison d’un Phénomène & des circonstances qui l’accompagnent, ni à priori, par le moyen des vérités que nous connaissons déja, ni a posteriori, par le secours des Expériences. (Amsterdam 1742)

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

Given this definition, Du Châtelet presents a sophisticated methodology of hypotheses, which substantiates the relevance and importance of hypotheses and allows to demarcate good from bad hypotheses. What role do hypotheses play? And which rules for their use must be observed? Here is Du Châtelet’s answer:

Hypotheses are useful because they can help us to discover new truths (InstPhy, § 58). For example, it would have been impossible to make many discoveries in astronomy without hypotheses. This is what happened with Saturn’s ring’s discovery by Huyghens. Consequently, hypotheses are an essential part of our «art d’inventer» (InstPhy, § 71), i.e., they have a heuristic function. Further, hypotheses promote unifying a theory by its extension. To mention an example which Du Châtelet gives: Kepler’s laws relating to the planetary ellipses were explained by Newton’s generalized laws of motion, i.e. Newton’s universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler’s Law of Harmonies relating to the planetary ellipses (InstPhy, § 58).

According to Du Châtelet there are at least two rules for the use of hypotheses: Firstly, hypotheses must neither contradict the principle of the sufficient reason nor contradict other fundamental principles of knowledge (see chapter 1). The rule says that, regardless of any confirmation by experience, not all hypotheses are acceptable but only ones that fit the principle of sufficient reason along with the other principles. In fact, a hypothesis should be at least possible and cannot be a “chimera” (InstPhy, § 68). Secondly, we should have reliable knowledge of facts and should know all the circumstances attendant upon the phenomena we intend to explain. A further rule, which is not explicitely mentioned as rule, is based on the asymmetry between verification and falsification of hypotheses (InstPhy, § 64):

One experiment is not enough for a hypothesis to be accepted, but a single one suffices to reject it when it is contrary to it. (Copyright © 2009 BZ)

Une expérience ne suffit pas pour admettre une hypothèse, mais une seule suffit pour la rejeter lorsqu’elle lui est contraire.

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

As example Du Châtelet mentions that Copernicus abandoned Ptolemy’s theory because the latter was disproved by astronomical phenomena. This continual process, made of invention of hypotheses, empirical test and correction of them, represents scientific progress as an evolving and open-ended process of trial and error. The method of trial and error can increase the probability of hypotheses and the method of falsification can help us approximate the truth by ruling out those hypotheses that cannot explain observed phenomena. This idea is usually called the problem of verisimilitude, or truthlikeness, which was deeply connected with the development of probability theory in the 18th century. A hypothesis is considered valid until it is not falsified. However, since just one new contrary experience can be enough to uncover its falsity, the truth degree of any hypothesis cannot reach further than likelihood or probability (InstPhy, § 67):

Hypotheses, then, are only probable propositions that have a greater or lesser degree of certainty, depending on whether they satisfy a more or less great number of circumstances attendant upon the phenomenon that one wants to explain by their means. And, as a very great degree of probability gains our assent, and has on us almost the same effect as certainty, hypotheses finally become truths when their probability increases to such a point that one can morally present them as a certainty. (Copyright © 2009 BZ)

Les hypotheses ne sont donc que des propositions probables qui ont un plus grand ou un moindre degré de certitude, selon qu’elles satisfont à un nombre plus ou moins grand des circonstances qui accompagnent les Phénomene que l’on veut expliquer par leur moyen; & comme un très-grand degré de probabilité entraîne notre assentiment, & fait sur nous presque le même effet que la certitude, les hypotheses deviennent enfin des verités, quand leur probabilité augmente à un tel point, qu’on peut la faire moralement passer pour une certitude.

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

It is probable that Du Châtelet’s rhetoric was influenced by Willem Jacob ’s Gravesande, who redefined Descartes’ division between moral certainty and probability on the one hand and Locke’s division between certainty and probability on the other in order to argue that Newtonian natural philosophy was based on moral evidence. And this moral evidence was to be regarded as just as persuasive as mathematical evidence. In his Introductio ad Philosophiam, which was translated into French in 1737, the year of the second Latin edition, ’s Gravesande worked out a methodology of hypotheses (“De usum hypothesim”) which contains six rules for the use of hypotheses. He also mentions Christiaan Huygens’s hypothesis concerning a ring around the planet Saturn.

However, Du Châtelet’s first rule cannot be found under s’ Gravesande list of rules. Du Châtelet’s first rule remembers more to Christian Wolff’s discussion of hypotheses, which was stimulated due to his correspondence with Leibniz during the last three years of Leibniz’s life. In “De hypothesibus philosophicis,”  published in Horae subsecivae Marburgenses (1729, § 5), Wolff used the same example as Du Châtelet did in order to compare the heuristic function of hypotheses in physics (astronomy) with that in mathematics (arithmetics) and to emphasize the crucial role of the trial-and-error method within the context of discovery (ars inveniendi). In his Philosophia Rationalis Sive Logica (1728, § 127), Wolff analyzed the use of hypotheses in arithmetic. For instance, division with a compound divisor on a Pythagorean abacus cannot reveal the true quotient. In this case an assumption is made, similar to a philosophical hypothesis, that the whole divisor is contained in the corresponding dividend as many times as the first number of the divisor is contained in the first number of the corresponding dividend. Compare Du Châtelet  (InstPhy, § 59):

It is the same with numbers. Division, for example, is founded on hypotheses only. Without hypotheses you could not divide, for when you begin division, you suppose that the divisor is contained in the dividend as many times as the first number of the divisor is contained in the first number, or in the first two numbers of the dividend; and then you verify this supposition by multiplying the divisor by the quotient, and by subtracting from the dividend the product of this multiplication. If you find that this subtraction cannot be done, you conclude that the quotient is too big, and then you correct it. This whole operation is done by means of hypotheses. (Copyright © 2009 BZ)

Les Hypothèses sont si nécessaires que sans elles on ne pourroit faire la plupart des opérations que l’on fait fur les nombres. La division, par exemple , n’est fondée que sur des hypothèses, & sans hypothèses, vous ne pourriez diviser; car lorsque vous commencez la division, vous supposez que le diviseur est contenu dans le dividende autant de fois que le premier chifre du diviseur est contenu dans leprémier chifre, ou dans les deux prémiers chifres du dividend; & alors vous vérifiez cette supposition en multipliant le diviseur par le quotient , & en sousstraiant du dividende le produit de cette multiplication. Si vous trouvez que cette soustraction ne peut se faire, vous conciuez que vous avez trop mis au quotient, & alors vous le corrigez, ainsi, toute cette opération se fait par le moyen des hypothèses. (Amsterdam 1742)

The attempt to “immunize” a theory against factual disconfirmations is often carried out by underpinning a hypothesis by an ad hoc hypothesis, i.e. a hypothesis which is made in order to save it from being falsified. Against this expedient, Du Châtelet argues that the consequencec that confirm a hypothesis should be direct and necessary, not intermediated by auxiliary assumptions (InstPhy, § 69):

So it is necessary not only that all one supposes be possible, but also that it be possible in the manner one uses it; and that the phenomena result necessarily, and without the obligation to make new suppositions. Otherwise, the supposition does not deserve the name of hypothesis; for a hypothesis is a supposition that explains a phenomenon. When the necessary consequences do not follow from it, and to explain the phenomenon, a new hypothesis must be created in order to use the first, this hypothesis is only a fiction unworthy of a philosopher. (Copyright © 2009 BZ)

Il est donc nécessaire non seulement que tout ce qu’on suppose soit possible, mais encore qu’il soit possible de la manière qu’on l’employe ; & que les Phénomenes en [de l’hypothèse] découlent nécessairement, et sans qu’on soit obligé de faire des suppositions nouvelles : sans cela, la supposition ne merite pas le nom d’hypothese; car une hypothese est une supposition qui rend raison d’un Phénomene. Or quand elle n’en rend point raison par des conséquences nécessaires, & qu’on est obligé de faire des hypotheses nouvelles pour faire usage de la premiere, ce n’est qu’une fiction indigne d’un Philosophe. (Amsterdam 1742)

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

Despite the general discredit fallen on the Cartesian hypothesis of whirls, Du Châtelet warns that at least one part of it can be kept. Huyghens had shown that in a Cartesian whirl bodies should fall perpendicularly to the earth axis, and not to its center. Therefore, a “tourbillon” of fluid matter, as it is supposed in Descartes’ hypothesis, could not bring about the phenomenon of falling bodies as we experience it. Similarly, Newton demonstrated that the Cartesian explanation of the planetary orbits relying on their transportation by fluid matter is incompatible with Kepler’s laws (InstPhys, § 65). However, for Du Châtelet it would be wrong to reject any theory of fluid matter or to conclude that vortices did not exist because of Descartes’ mistakes. One part of his hypothesis might be saved suggesting an explanatory theory of Newton’s attraction in terms of the existence of a fluid ether that pervades the whole and works as a vehicle for the force of attraction (InstPhys, § 397-398). Insofar the hypothesis of a force acting at a distance could be avoided. The crucial point is that Newton’s theory explains one and the same phenomena (e.g. tide) on the basis of less unconfirmed premises.

At the end of the chapter, Du Châtelet shows that “good hypotheses were always made up by the greatest men” (InstPhy, § 71):

Therefore, the good hypotheses will always be the work of the greatest men. Copernicus, Kepler, Huygens, Descartes, Leibniz, M. Newton himself, have all imagined useful hypotheses to explain complicated and difficult phenomena; and the examples of these great men and their success must show how much those who want to banish hypotheses from philosophy misunderstand the interests of the sciences. (Copyright © 2009 BZ)

Les bonnes hypotheses seront donc toujours l’ouvrage des plus grands hommes. Copernic, Képler, Hughens, Descartes, Leibnits, M. Newton lui même, ont tous imagine des hypotheses utiles pour expliquer des Phénomenes compliqués & difficiles; & les exemples de ces grands hommes & leur succès doivent nous faire voir combien ceux qui veulent banner les hypotheses de la Philosophie, entendent mal les intérêts des sciences. (Amsterdam 1742)

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

The transition from the Ptolemaic system to Copernicus’ system is described as the passage from a hypothesis that is contradicted by astronomical observations to one more exact and truer to the data. Also the transition from Galileo’s circular orbits of the planets to the elliptical ones, hypothesized by Kepler, until Newton’s explanation of the orbits in terms of centripetal and centrifugal forces is characterized by Du Châtelet as a progress from one less to one more performing hypothesis (InstPhy, § 58). The utility and even indispensability of hypotheses is vividly described in this historical recours.

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See also the entry HYPOTHESE, s. f. (Métaphysiq.) in d’Alembert’s and Diderot’s Encyclopédie.

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