## Crucial Instances in the Principia

*Kirsten Walsh writes…*

In the General Scholium, which concluded later editions of *Principia*, Newton described the work as ‘experimental philosophy’:

In this experimental philosophy, propositions are deduced from phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies, and the laws of motion and the law of gravity have been found by this method.

On this blog, I have argued that we should take this statement at face value. In support, I have emphasised similarities between Newton’s work in optics and mechanics. For example, I have considered the kind of evidence provided in each work, arguing that both the *Principia*’s ‘phenomena’ and the *Opticks*’s ‘experiments’ are idealisations based on observation, and that they perform the same function: isolating *explananda*. I have also emphasised Newton’s preoccupation in the *Principia* with establishing his principles empirically. Finally, I have suggested that this concern with experimental philosophy, in combination with his use of mathematics, made Newton’s method unique.

In my last blog post, I wondered if we should regard Newton’s methodology as an extension of the Baconian experimental method, or as something more unique. I have written many blog posts discussing the Baconian aspects of Newton’s optical work (for example, here, here and here), but the Baconian aspects of the *Principia* are less well-established. I can identify at least three possible candidates for Baconianism in the *Principia*. The first, suggested by Daniel Schwartz in recent conversation, is that book 3 contains what might be interpreted as Baconian ‘crucial instances’. The second, discussed by Steffen Ducheyne, is that Newton’s argument for universal gravitation resembles Bacon’s method of induction. The third, discussed by Mary Domski, is that the mathematical method employed in the *Principia* should be viewed as part of the seventeenth-century Baconian tradition. In this post, I’ll focus on Schwartz’s suggestion—the possibility there is a crucial instance in book 3 of the *Principia*—I’ll address the rest in future posts.

To begin, what is a ‘crucial instance’? For Bacon, crucial instances (*instantiae crucis*) were a subset of ‘instances with special powers’ (ISPs). When constructing a Baconian natural history, ISPs were experiments, procedures, and instruments that were held to be particularly informative or illuminative of aspects of the inquiry. These served a variety of purposes. Some functioned as ‘core experiments’, introduced at the very beginning of a natural history, and serving as the basis for further experiments. Others played a role later in the process. This included experiments that were supposed to be especially representative of a certain class of experiments, tools and experimental procedures that provided interesting investigative shortcuts, and model examples that came close to providing theoretical generalisations.

Crucial instances are part of a subset of ISPs that were supposed to aid the intellect by “warning against false forms or causes”. When two possible explanations seemed equally good, then the crucial instance was employed to decide between them. To this end, it performed two functions: the negative function was to eliminate all possible explanations except the correct one; the positive function was to affirm the correct explanation.

According to Claudia Dumitru, Bacon’s crucial instances have a clear structure:

- Specify the
*explanandum*; - Consider the competing explanations (these are assumed to exhaust the possibilities);
- Derive a consequence from one explanation that is incompatible with the other explanation(s);
- Test that consequence.

Are there any arguments in the *Principia* that look like crucial instances? I think there’s at least one: Newton’s famous ‘Moon test’. Let’s have a look at it.

In proposition 4 book 3, Newton used his Moon test to establish that “*The moon gravitates toward the earth and by the force of gravity is always drawn back from rectilinear motion and kept in its orbit*”. Here, Newton argued that the inverse-square centripetal force, keeping the moon in orbit around the Earth, is the same force that, say, makes an apple fall to the ground, namely, gravity. I think we can tease out the features of a Baconian crucial instance from Newton’s reasoning here.

Firstly, there is an *explanandum*: what kind of force keeps the Moon in its orbit and prevents it from flying off into space? Secondly, two possible explanations are provided: the force is either (a) the same force that that acts on terrestrial objects, namely, gravity; or (b) a different force. Thirdly, we have a consequence of (a) that is incompatible with (b): if the moon were deprived of rectilinear motion, and allowed to fall towards Earth, it would begin falling at the rate of 15 ^{1}/_{12} Paris feet in the space of one minute, accelerating so that at the Earth’s surface it would fall 15 ^{1}/_{12} Paris feet in a second. Finally, we see a test of that consequence: the calculations based on the size and motion of the Moon, and its distance from the Earth. The results are taken to support (a) and refute (b).

I have three concluding remarks to make.

Firstly, interpreting the Moon test as a crucial instance involves ‘rational reconstruction’. In the text, Newton starts by calculating the rate at which the Moon would fall, and shows that this supports proposition 4. But I think my reading of this as a crucial instance is supported by Newton’s concluding remarks:

For if gravity were different from this force, then bodies making for the earth by both forces acting together would descend twice as fast, and in the space of one second would by falling describe 30

^{1}/_{6}Paris feet, entirely contrary to experience.

Here, Newton described the Moon test as a crucial instance: he used an observation to choose between two competing explanations of the *explanandum*.

Secondly, when looking for crucial instances in the *Principia*, it might be tempting to start with the phenomena, listed at the beginning of book 3. Elsewhere, I have argued that these resemble Newton’s experiments in the *Opticks*, which function as instances with special powers. But the label ‘crucial instance’ describes the function, not the content, of an empirical claim. And so, to see if they provide crucial instances, we need to consider how the phenomena are used. In fact, I think they do provide crucial instances for Newton’s rejection of Cartesian vortex theory in favour of universal gravitation, found at the end of book 2. But again, this requires rational reconstruction.

Finally, there is the issue of historical influence. I have shown that Newton employed the Moon test to decide between two competing explanations, and that this argument resembles one of Bacon’s crucial instances. However, one might think that this was simply a good approach to empirical support, and that Newton was using his common-sense. So perhaps we shouldn’t take this to indicate (direct or indirect) influence. And so I have a question for our readers: was this style of reasoning uniquely Baconian?