Do Newton’s rules of reasoning guarantee truth … must they?

Abstract

Newton’s Principia introduces four rules of reasoning for natural philosophy. Although useful, there is a concern about whether Newton’s rules guarantee truth. After redirecting the discussion from truth to validity, I show that these rules are valid insofar as they fulfill Goodman’s criteria for inductive rules and Newton’s own methodological program of experimental philosophy; provided that cross-checks are used prior to applications of rule 4 and immediately after applications of rule 2 the following activities are pursued: (1) research addressing observations that systematically deviate from theoretical idealizations and (2) applications of theory that safeguard ongoing research from proceeding down a garden path.

1. Introduction

Book 3 of Isaac Newton’s Mathematical principles of natural philosophy begins by introducing four rules of reasoning for natural philosophy.1 As will be shown, these rules allow Newton to make valuable inductions in the early propositions of Book 3, inductions that could not have been made without rules of this sort. Nevertheless, there is a concern about whether Newton’s rules guarantee truth. I intend to show that this concern is misguided. However, against the criticism that these rules do not guarantee valid inferences, Newton’s rules are defendable insofar as they satisfy Goodman’s criteria for inductive rules and Newton’s own methodological program of experimental philosophy; provided that cross-checks are used prior to applications of rule 4 and immediately after applications of rule 2 the following activities are pursued: (1) research addressing observations that systematically deviate from theoretical idealizations and (2) applications of theory that safeguard ongoing research from proceeding down a garden path. This paper will not include a defense of Goodman’s criteria, but will argue that if his criteria are accepted, then given my defense of Newton’s rules as satisfying that criteria, Newton’s rules of reasoning are valid.

2. Newton’s rules of reasoning

Newton made numerous drafts of his rules before including them in the Principia.2 Newton’s care in stating these rules and including only certain ones in subsequent editions suggests that each rule is important. So I will dedicate some space to presenting each rule and discussing its importance to the Principia.

Below are Newton’s first and second rules of reasoning from the third edition of the Principia:

Rule 1. No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena. (Newton, 1999, p. 794)

Rule 2. Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same. (Newton, 1999, p. 795)

Rule 1 instructs us to explicate 3 nature in the simplest way possible while still maintaining truth. Although Newton mentions the philosophic adage that ‘nature does nothing in vain’, given Newton’s distaste for ‘hypotheses’,4 Newton’s real intent for rule 1 is to limit claims about differences in causes to ones that the empirical data force (Smith, 2002b, p. 160). In other words, rule 1 restricts theory making to information that has been gathered through experiments or other empirical observations even if only indirectly.5

Given the ‘therefore’ and the placement of rule 2 almost immediately after rule 1, rule 2 is more likely the consequent of rule 1 than a separate rule all by itself (Harper, 2002, p. 183). So, given rule 1, rule 2 assigns the same cause to the same phenomena. Newton first uses rules 1 and 2 in proposition 4 (prop. 4) for the moon test, which equates terrestrial gravity with lunar centripetal acceleration.6 (See Appendix A for a thorough discussion of the moon test.) Newton then uses rules 1 and 2 in prop. 5 to attribute the five primary planets’ centripetal acceleration to gravity.7 Newton uses the rules a final time in the scholium to prop. 5 to equate all celestial centripetal acceleration with gravity.

The importance of rules 1 and 2 in their first application is that without them Newton cannot infer that the moon’s acceleration toward the earth and terrestrial bodies’ acceleration toward the earth are in fact from identical causes. Although Newton has deduced that the value for lunar centripetal acceleration is very nearly equal to the value for terrestrial gravity—since Newton’s value in prop. 4 falls within the error bounds of Christiaan Huygens’s precise measurement of terrestrial gravity—the most that Newton has proven is that these accelerations have corresponding magnitudes and direction. Newton needs an inductive rule to instruct him that it is permissible to conclude from this similarity that these two phenomena are due to the same cause. Although rules 1 and 2 are not necessary to make this induction, they suffice.

The second application of rules 1 and 2 serves a similar, but slightly different purpose. In prop. 5, although Newton has established in prop. 2 that the primary planets have centripetal accelerations with the same properties as the moon—namely, that they are proportional to the inverse-square of the distance from their centres of orbit in which equal areas are swept out in equal times and that they are directed toward their centres of orbit—without an inductive rule to permit an identity relation between the moon’s kind of acceleration and the kind keeping the primary planets in orbit, Newton has only shown that these accelerations have the same properties.8 Invoking rules 1 and 2 classifies each of these accelerations as the same kind. Rules 1 and 2 are used in the same way to designate all celestial centripetal acceleration as the gravitational kind. So, in all three applications the rules invoke identical causes; however, in the first application the cause is a particular and in the last two the cause is a kind.9

The following is Newton’s third rule of reasoning:

Rule 3. Those qualities of bodies that cannot be intended and remitted [that is, qualities that cannot be increased and diminished] and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally. (Newton, 1999, p. 795)

William Harper refers to rule 3 as Newton’s ‘argument from phenomena’ (Harper, 2002, p. 175), and this rule does just that. If a property satisfies a certain empirical criterion, rule 3 instructs us to consider that property as universal, where the criterion is being a constant property of all bodies that are observable through experiment.

Newton explicitly invokes rule 3 only once, although he implicitly invokes it two more times. In corollary 2 of prop. 6, Newton uses rule 3—along with the results from his pendulum experiments that show a body’s weight to its mass is a constant at the earth’s surface—to make the induction that all bodies gravitate toward the earth.10 However, since prop. 6 claims to have shown that ‘all bodies gravitate toward each of the planets’, Newton must have implicitly invoked rule 3 a second time.11 Probably, Newton took the generalization for the earth and rule 3 to show that prop. 6 holds.12 Also, in prop. 7, although Newton shows that all of the particles of any two planets gravitate toward each other, he must have implicitly invoked rule 3 in order to extend the argument to all of the particles in the universe, which as William Harper notes (2002, p. 192), could have been done since Newton’s proof included all of the planets in reach of experiments at that time.13

One importance of rule 3 is that without it Newton cannot assert that all bodies near planets gravitate toward those planets in the same fashion as observed bodies near planets (for example, satellites and terrestrial bodies).14 However, establishing the latter is not crucial to Newton’s argument for gravity between particles since gravity between planets can be deduced from celestial centripetal acceleration being due to gravity (which I will explain later). Another importance of the rule is that the last step in the argument for universal gravity requires a generalization to all particles in the universe that only rule 3 provides. Also, notice that the first use of rule 3 is for an induction from terrestrial bodies to all bodies near the earth while the second use is for an induction from all bodies near the earth to all bodies near any planet. Thus the second induction is nested, where being ‘nested’ is having the status of depending on other inductions.15 In effect, each successive application of rules 1, 2, and 3 lead to nested inductions.16 ^, 17

The fourth rule of reasoning is below:

Rule 4. In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions. (Newton, 1999, p. 796)

Coined ‘the rule of conduct [for] empiricism’ by Alexandre Koyré (1965, p. 268), rule 4 does three things for natural philosophy. First, it instructs us that theories deduced from phenomena and generalized by induction always override theories that fail to do so. Second, the phrase ‘should be considered exactly or very nearly true’ instructs us to model the world as idealizations.18 Third, the clause after the last comma suggests that natural philosophy is an ongoing program of making ever better idealizations of the world. The latter two points turn natural philosophy into an exact science, not insofar as it calls for idealizations, but insofar as it suggests that phenomenal deviations from those idealizations, or theory dependent second-order phenomena,19 must be addressed in a new idealization that better approximates the world (Smith, 2002b, pp. 157–160).

Newton invokes rule 4 only once although he could have invoked it several times. In the scholium to prop. 5, Newton uses rule 4 after applying rules 1 and 2.20 The effect of applying the rule here is to establish it as a principle that centripetal acceleration for celestial bodies is the same kind of acceleration as terrestrial gravity. Newton needs the latter to be a principle in order to establish universal gravity in a way that is not hypothetico-deductive.21

As we see from prop. 7, Newton’s argument for gravity between particles depends as much on his laws of motion as it does on the attraction between planets being due to gravity, since gravity between planets is a premise in the argument:

We have already proved that all planets are heavy [or gravitate] toward one another and also that the gravity toward any one planet, taken by itself, is inversely as the square of the distance of places from the center of the planet. (Newton, 1999, p. 810)

However, in order to establish gravity between planets as a principle, it has to follow from some other principle. As it happens, gravity between planets follows from celestial centripetal acceleration being one in kind with gravity. Newton provides the proof in corol. 1, prop. 5:

Therefore, there is gravity toward all planets universally. For no one doubts that Venus, Mercury, and the rest [of the planets, primary and secondary,] are bodies of the same kind as Jupiter and Saturn. And since, by the third law of motion, every attraction is mutual, Jupiter will gravitate toward all its satellites, Saturn toward all its satellites, and the earth will gravitate toward the moon, and the sun toward all the primary planets. (Newton, 1999, p. 806)

So, establishing it as a principle that gravity is the kind of acceleration keeping celestial bodies in orbit is important.

Although he did not, Newton could have invoked rule 4 after each induction, since this would have made each induction exactly or very nearly true. Newton certainly does (implicitly) invoke rule 4 after prop. 7, since in prop. 8 through prop. 13 he uses the theory of universal gravity as if it holds exactly in order to demonstrate that Copernicanism is essentially correct (Newton, 1999, pp. 811–819).

3. Do Newton’s rules guarantee truth … must they?

The easiest case to make against Newton’s rules is against the inference that rule 2 allows. Rule 2 bluntly instructs us to assign the same cause to the same phenomena. Newton gives the following examples of how to apply rule 2:

Examples are the cause of respiration in man and beast, or of the falling of stones in Europe and America, or of the light of a kitchen fire and the sun, or of the reflection of light on our earth and the planets. (Newton, 1999, p. 795)

However, the recent epidemic of Severe Acute Respiratory Syndrome (SARS) serves as a glaring counterexample to rule 2. SARS induces the same symptoms as the common cold or influenza, but its cause is a different virus (as shown by its recalcitrance to common cold or influenza treatments).22 In that case, rule 2 allows for a false inference when applied to SARS and either the common cold or influenza.

The SARS argument against Newton’s rule would be of concern if the criticism were not misguided. The argument against Newton’s rule inappropriately requires truth from inductive rules instructing induction from phenomena.

Nelson Goodman identifies the old problem of induction for induction from phenomena as the difficulty of devising rules such that we know their projections will turn out to be true. Goodman replies that no set of rules would fulfill this criterion since it requires knowledge in a situation where we admittedly do not have the sought after knowledge. If we had knowledge of true projections beforehand, then we would have no need for rules that project. Instead, Goodman argues that a set of ‘inductive rules’ 23 need only be valid. Furthermore, for Goodman (1979, p. 64), since inferences are rejected if they violate rules we are unwilling to amend and rules are amended if they yield inferences we are unwilling to accept, a set of inductive rules are valid if …

• (1.1)No rule in the set yields an inference that we are unwilling to accept.
• (1.2)Either ‘inferences we are unwilling to accept’ or ‘rules we are unwilling to amend’ is defined.

Criterion (1.1) follows from Goodman’s condition for amending rules. Criterion (1.2) is needed because (1.1) depends on either Goodman’s condition for rejecting inferences and a definition for ‘unamendable rules’ or a definition for ‘unacceptable inferences’. Since empty sets, sets of only trivial rules, and sets with conflicting unamendable rules and unacceptable inferences are possible from (1.1) and (1.2), I will add the following to Goodman’s criteria:

• (1.3)There is at least one rule in the set.
• (1.4)All inferences that we are unwilling to do without follow from some rule in the set.
• (1.5)No inference that we are unwilling to accept follows from a rule that we are unwilling to amend.24
• (1.6)‘Inferences that we are unwilling to do without’ is defined.

Also realize that given the sheer complexity of orbital motion, Newton was well aware of the difficulty in capturing the phenomenon by exact mathematical laws. This is good reason to think that Newton was aware of the fact that his rules of reasoning could lead to false inferences even before he wrote them. In a 1684 tract entitled The motion of bodies in orbit, Newton expresses his opinion that orbital motion is exceedingly complex:

By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. There are as many orbits of a planet as it has revolutions, as in the motion of the Moon … But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind. (Newton, 1962, p. 281)

Although the initial criticism has been addressed, an immediate concern arises about Newton’s rules. Since inductive rules need only be valid, are Newton’s rules valid?

4. Are Newton’s rules valid?

The best criticism against the validity of rule 1 is that it pursues simplicity, which is unrelated to scientific inquiry, even on an empiricist’s account. Harper notes that some empiricists define empirical success entirely indifferent to simplicity (Harper, 2002, p. 185). Huygens provides a few of these definitions in his Treatise on light when he says that we best approximate the natural world when theories match observed phenomena and when theories predict new phenomena (Huygens, 1989). So, explaining or predicting phenomena should be the appropriate measures for empirical success regardless of whether those explanations or predictions are simple.

Notice the style of argument of this criticism. It argues that if phenomena are explained or predicted, then we have empirical success. It does not restrict empirical success to these criteria. In effect, we could accomplish empirical success by other criteria, even in the absence of prediction or explanation. I am not proposing that Newton thought empirical success is indifferent to prediction or explanation or that none of Newton’s inspiration for this rule was due to a belief that nature is essentially simple. I am suggesting that Newton had a valid empirical reason to include this rule besides a mere appeal to simplicity. Given Newton’s intent of establishing experimental philosophy as a new way of doing natural philosophy, this rule should be viewed as an instruction to limit theories to the empirical data supporting them and to discourage ‘hypotheses’.25 Newton’s opinion of experimental philosophy is clear from the penultimate paragraph of the General Scholium:

I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. (Newton, 1999, p. 943)

Newton’s views agree with this defense of rule 1 as we see from his comments in the first paragraph after rule 3:

Certainly idle fancies ought not to be fabricated recklessly against the evidence of experiments, nor should we depart from the analogy of nature, since nature is always simple and ever consonant with itself. (Ibid., p. 795)

This explanation of rule 1 becomes even more plausible once we note the difficulty that seventeenth-century philosophers had with marshalling evidence for a phenomenon that supported only one explanation of that phenomenon. For instance, data that supported the Copernican system for the planets also supported the Tychonic system (Gingerich, 1993, pp. 30–31).26 In that case, it makes sense to devise a rule that restricts theory construction to only those propositions that are supported by empirical evidence (since the task is difficult enough even before ‘hypotheses’ are proposed). Having a rule that does this is the first step toward winnowing one overriding explanation for a phenomenon out of a consortium of possible explanations. Therefore, despite Newton’s preoccupation with simplicity, rule 1 is justified by its importance to the methodology of Newton’s experimental philosophy.

The validity of rule 2 is questionable in the following methodological sense: beyond the fact that the rule occasionally allows for false inferences is the fact that the rule systematically misleads under certain specifiable circumstances. For instance, when the empirical data supporting the phenomena are inadequate or when the phenomena are exceedingly complex, rule 2 should be abandoned. Abandoning the rule under these circumstances serves to safeguard a theory from proceeding down a garden path,27 perhaps involving fictional physical kinds (since assigning phenomena to the same cause designates the cause as a kind). Furthermore, given the complexity of planetary motion or the fact that the empirical data were not precise enough in the derivation of universal gravity,28 while rule 2 may be justifiable in other circumstances, certainly it is not justifiable for Newton’s purposes. As a result, rule 2 is invalid not because it is misleading under some circumstances, but because it is misleading under the very circumstances for which Newton intends to use it.

First, the concern over the applicability of rule 2 should not depend on the complexity of the phenomena. One reason why the rule is useful is because it allows for an idealization that is testable with theory-mediated measurements,29 which is a method of handling complex phenomena. For example, since the first application of rule 2 leads to the idealization that lunar centripetal acceleration is due to the same cause as terrestrial gravity, and since from prop. 3 lunar centripetal acceleration varies as the inverse-square of the distance from the earth’s centre, the theory mediates the measurement that terrestrial gravity varies as the inverse-square of the distance from the earth’s centre.30 ^, 31 With this new measurement for terrestrial gravity, experiments can be done to test whether terrestrial gravity actually varies in the way that the theory prescribes. Varying values for terrestrial gravity may be predicted in the following way.

Assuming that the earth is spherical, terrestrial gravity is proportional to the inverse-square of the radius to the earth’s surface. This makes the ratio of terrestrial gravity at some height h from the earth’s surface, g′, to terrestrial gravity at the earth’s surface, g, equal to the ratio of the square of the earth’s radius r to the square of the quantity of r plus h.

g'/g=r2/(r+h)2 (2.1)

Substituting the product of r and a fraction k for the variable h, executing some operations, then re-substituting the quotient of h and r for k, provides a function for g′ dependent on r. But since the circumference of the lunar orbit C was available in the seventeenth century (Newton, 1999, p. 804) and the distance between the earth and moon’s centres was estimated to be 60r (ibid., pp. 803–804), a value for r could have been calculated. Substituting the value for r, we find a simple function for g′ dependent only on h.

g'=[1/(120πh/C+1)2]g (2.2)

So, from (2.2), depending on how precise our measurement of g is, we can drop objects over finite distances at some h in order to detect a discrepancy between g and g′; where precision for g is inversely related to the minimum h required for detecting a discrepancy between g and g′.32

The worry about following a garden path due to complex phenomena may be addressed with cross-checks,33 or in other words, some method for checking the inductive inferences made by the theory (for example, by deriving the same inference using different sets of assumptions).34 So, for these two reasons, rule 2 should be viewed as a pragmatic tool for handling complex phenomena. However, the second point is more worrisome.

Since the empirical data Newton had available were not precise enough to rule out problematic alternative derivations—like celestial centripetal acceleration varying as the distance (see note 28)—the point is well taken about an application of rule 2 being systematically misleading. However, this criticism does not invalidate rule 2, it only cautions the application of the rule under this circumstance. Nevertheless, admitting that rule 2 systematically misleads under the very circumstance that Newton uses it is an issue that should be addressed. But let me put this issue aside for the moment in order to address the validity of the remaining two rules.

As for rule 3, its validity should not be attacked on the basis that it embodies the new riddle of induction,35 which is the difficulty of describing valid projection. Since this riddle permeates any inductive rule, the best way to question the validity of rule 3 is to question unique aspects of it. In Newton’s day, explicitly including the phrase ‘on which experiments can be made’ was unique (Newton, 1999, p. 795). However, immediately the question arises: what counts as an experiment? Peculiarly enough, nowhere in the Principia does Newton discuss the parameters of an acceptable experiment, which seems odd since Newton is promoting experimental philosophy.36 In order to capture the strength of this criticism, let me use an analogy. Instead of its actual wording, imagine rule 3 stated, ‘on which blibbles can be made’. However, this instruction only raises the question: what is a blibble? In that case, questioning what counts as an experiment is not absurd when questioning the validity of rule 3.

Although I agree that a vague rule is invalid, it is not clear that Newton’s audience cannot extrapolate the parameters of an acceptable experiment by reading the Principia. Examples are the pendulum experiments Newton presents in the scholium to the laws of motion and prop. 6 of Book 3. However, from reading prop. 6, we see a new type of experiment that Newton is willing to count as acceptable: vicarious investigations.37 In Newton’s satellite test, he uses theory-mediated measurements to investigate whether the sun emits an equal gravitational field on Jupiter and its satellites.

Newton begins by assuming that at some distance r from the sun, the ratio of one of Jupiter’s satellite’s weight to mass is greater than the ratio of Jupiter’s weight to mass. He designates the first ratio as the acceleration d and the second as the acceleration e.

Ws/ms>WJ/mJ (3.1)

d=Ws/ms (3.2)

e=WJ/mJ (3.3)

Using the theory-mediated measure for the force of gravity and since weight is the force of gravity, weight is proportional to the inverse square of the distance. Rearranging some terms shows that the ratio of the separation between Jupiter’s satellite and the sun to the separation between Jupiter and the sun is as the square root of d to the square root of e.

d:e::1/rsS2:1/rJS2 (3.4)

d:e::rJS2:rsS2 (3.5)

√d:√e::rJS:rsS (3.6)

Then Newton supposes that at some distance Jupiter’s satellite’s acceleration is greater than Jupiter’s by one thousandth of Jupiter’s. (The point here is to suppose a miniscule difference in accelerations.)

d=(1/1000)e+e (3.7)

In that case, the separation between Jupiter’s satellite and Jupiter’s distances from the sun would be two thousandths of the separation between Jupiter and the sun.

rJS-rsS=rJS-(√1000/√1001) rJS≈(1/2000)rJS (3.8)

Since this value is a fifth of the distance between Jupiter and its outermost satellite, it would be easily observed. Since such a discrepancy has not been observed, the sun’s gravity must be equal at equal distances from the sun. In other words, the ratio of weight (as a function of distance) to mass is a constant.38

Wr/m=constant (3.9)

Since the satellite test provides a constant quality of bodies, if the satellite test exhausts the vicarious investigations that can be made, we could use rule 3 to make an inductive inference.39 Newton’s rule 3 is valid because the Principia provides clear examples as to what counts as an acceptable experiment.

Looking at rule 4, we see a glaring problem with respect to its validity. Newton employs the word ‘phenomena’ twice in order to refer to the standard on which we should base our inductions. However, as I have already mentioned, ‘phenomena’ as Newton uses it refer to generalizations from empirical observations, not empirical observations themselves (see note 5). Examples are the six propositions that comprise Newton’s ‘Phenomena’ chapter in Book 3. The question now becomes whether phenomena are deduced or induced from the data. If phenomena are induced, then rule 4 is nested on an induction whose rule for construction has not been explicitly provided.

The best way to determine if phenomena are induced and if so, what rule is used to construct them, is to step through one of Newton’s phenomena constructions. Phenomenon 1 is below:

Phenomenon 1. The circumjovial planets [or satellites of Jupiter], by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being at rest—are as the ³/₂ powers of their distances from that center. (Newton, 1999, p. 797)

Newton presents data collected from Borelli, Townly, and Cassini representing the periods of Jupiter’s four known satellites and their distances from the centre of Jupiter. Each satellite’s period and mean distance is in Table 1 :

Table 1.

The periods and mean distances of Jupiter’s satellites as of 1726

Satellite	Period (in days)	Mean distance (in Jupiter radii)
1	1.769	5.46
2	3.551	8.61
3	7.155	13.71
4	16.689	24.42

If a number x with exponent y equals another number z, then the logarithm of z to base x equals y. Since the logarithm of z to base x equals the quotient of the logarithm of z to base ten divided by the logarithm of x to base ten, y equals that quotient. Finding the exponential relation between the satellites’ distances to their periods merely requires that we represent periods T by z and distances r by x. A manipulation shows that y is the slope m of the logarithm of the period to base ten as a function of the logarithm of the distance to base ten.

log T=m log r (4)

Since the equation is linear, the slope is found by finding the quotient of differences in log T divided by differences in log r. Three such slopes are found for each of the three intervals. Their values are in Table 2 . 40

Table 2.

Slopes for intervals of log T vs. log r

Interval	Slope
1	1.48
2	1.44
3	1.38

From Table 2 we see that the mean slope is 1.43, which is 0.07 from three halves. From this close approximation, Newton concludes that the periods of Jupiter’s satellites are as the three halves power of their distances.

However, an induction is needed to conclude the three halves power relation since there are higher-order values for the power of the distances that are closer to 1.43 (Harper, 2002, p. 196).41 For example, seven fifths is 0.03 from 1.43 and ten sevenths is only 0.00143 from the value. Furthermore, since Newton must have made an induction, in order to validate the induction, he must have used either an already mentioned inductive rule or an unmentioned inductive rule. An appeal to rule 1 does not work since seven fifths, ten sevenths, and three halves are all simple fractions and are all equally supported by the data (since the imprecision in data exceeds these fractions). An appeal to rule 2 or rule 3 is unhelpful since they do not apply. Invoking rule 4 is unhelpful because this rule presupposes that the induction has already been made. Hence, unless Newton is invoking a rule that he has not mentioned, his phenomena are invalidly inferred. Nesting a rule on an invalid induction makes the rule invalid. But, could Newton have had a valid reason to make an induction to the three halves power rule even if no inductive rule applied?

Given the difficulty of marshalling evidence sufficient to infer phenomena, Newton could have chosen the three halves power rule for the relation of periods to distances provided that the part of his theory developed so far could produce a sufficient cross-check of the relation. Newton’s theory before Book 3 can do such a cross-check, and the results of the cross-check support a three-halves power relation and fail to support a seven-fifths or a ten-sevenths power relation.

Corollary 7, prop. 4, Book 1 provides the following universal measure for centripetal ‘accelerative force’42 from the power rule of a body’s period to distance:

T∝rsiff Fa∝r1-2s (5.1)

Now, (5.1) can be used to measure the competing centripetal accelerative forces from the competing power rules for celestial periods to distances.

T∝r3/2iff Fa∝1/r2 (5.2)

T∝7/5iff Fa∝1/r12/5 (5.3)

T∝r10/7iff Fa∝1/r17/7 (5.4)

The cross-check arises from definition 7 when Newton defines centripetal accelerative force as varying directly with change in velocity and inversely with change in time (Newton, 1999, p. 407). Equating these different measures of the same force we find the following relation:

1/rq∝Δv/Δt (5.5)

Since celestial orbits are observed to be very nearly circular, we may approximate change in velocity over change in time by the circumference of a circle divided by the square of the period. With a few manipulations, the following proportion arises:

T∝r(1+q)/2 (5.6)

Substituting our competing values for the inverse 1 – 2s of the distance for q serves to check the relation of period to distance, since (5.1), (5.6) rest on different assumptions. The following shows the results of the check:

T∝r3/2if q=2 (5.7)

T∝r17/10if q=12/5 (5.8)

T∝r24/7if q=17/7 (5.9)

Since the relation in (5.7) is equal to the relation in (5.2), but the relations in (5.8) and (5.9) are not equal to the relations in (5.3) and (5.4), respectively, the check shows that the three halves power rule is the valid induction to make.

Since the cross-check assumes celestial orbits are circular, the check does not rule out higher-order power rules that better characterize less circular orbits. In this sense, the reliability of the check depends on the reliability of observational measurements that celestial orbits are at least very nearly circular. If Newton had that sort of data, then his induction to the three halves power rule in phen. 1 was valid. Similar arguments follow for Newton’s construction of phen. 2 through phen. 6. So, nesting rule 4 does not have to be invalid, since rule 1 could apply or other valid reasons for the induction on which rule 4 is nested could be proposed (for example, if a proposition passes a cross-check).43

5. Validating rule 2

Although the other rules have been validated, still there remains the lingering issue that challenges the validity of rule 2, namely, that the rule can mislead if the data are not precise enough to dismiss problematic alternative derivations. Although the explanation is complex, there are two reasons why rule 2 is valid despite the fact that it misleads in this way.

First, if the theory constructed in part by rule 2 proves successful in ongoing research, then the justification for rule 2 increases directly with that success. Since, as mentioned before, rule 2 allows for idealizations, explicating identified theory-dependent second-order phenomena with new idealizations lends increasing support to rule 2, since any new idealization depends in part on rule 2 if the original idealization depends in part on rule 2. Also, the process of developing increasingly better idealizations of the world reduces the amount of adjustments that a theory will need to make in order to explicate the world. In this sense, each successful explication of theory-dependent second-order phenomena with theory developed in part by rule 2 not only increases support for rule 2, but decreases the number of instances in which rule 2 can mislead. So, developing new idealizations and reducing the amount of future explications for theory-dependent second-order phenomena effectively increase the validity of rule 2 (Smith, 2002b, pp. 161–162). In Book 3, Newton’s attempt to explicate the inequalities in lunar motion is an example of how to validate rule 2 in this way (Newton, 1999, pp. 832–874).

Second, applying the theory developed in part from rule 2 to solve problems that could expose limitations of the theory is a safeguard that reduces the risk involved with rule 2. Said otherwise, relentlessly using a theory as soon as it is developed exposes the risky features of the theory (Smith, 2002b, p. 165). For example, since variation of terrestrial gravity is developed in part from rule 2—see my derivation of (2.2)—then immediately engaging in experiments to confirm the variation of terrestrial gravity safeguards our use of the rule.

Newton demonstrates the theory application safeguard by solving six problems presented after his work on lunar motion (Newton, 1999, pp. 874–938). For example, in problems 21 and 22 Newton applies gravitational theory to account for the trajectory of comets. In particular, Newton uses the theory to account for the periodicity of Halley’s comet, which recurs every seventy-five to seventy-seven years. Since the third edition of the Principia was published in 1726, and the last sighting of Halley’s comet before the third edition was in 1682, this particular safeguard could have been substantiated as early as thirty-one years after the Principia’s publication; which could have reduced the length of a garden path involving rule 2 to a matter of three decades. In that case, as long as theory application safeguards are pursued immediately after theory construction, then in a theory depending in part on rule 2, the length of a garden path involving rule 2 should be appreciably reduced.

These theory application safeguards should also be narrowly tailored to expose the risk arising from rule 2, since rules 1 and 3 have risk as well. However, since rule 1 always accompanies rule 2 in Book 3, the safeguard should be narrowly tailored to expose the risk involved in rules 1 and 2, but to exclude the risk involved in rule 3. Confirming the variation of terrestrial gravity is a good example of a narrowly tailored safeguard that does so. Also, notice that my argument for the validity of rule 2 is forward looking, insofar as the justification of the rule comes after the rule has been applied. But this is why explicating theory-dependent second-order phenomena and initiating theory application safeguards must be done immediately after theory construction. Successful ongoing research validates rule 2 only if the risk of using the rule propels ongoing research.

6. Conclusion

Earlier, plausibility for inductions from phenomena was restricted to the validity of those inductions; where validity was established with Goodman’s criteria.44 Addressing the best criticisms against the validity of each rule of reasoning serves to demonstrate that no rule yields an inference that we are unwilling to accept, where ‘unacceptable inferences’ was defined in each criticism. Since Goodman’s criteria are fulfilled, each rule of reasoning is valid—provided that the appropriate supporting data are available for rule 4 as well as explications for theory-dependent second-order phenomena and theory application safeguards are pursued for rule 2. Validating Newton’s rules does not mean that these are the only valid rules for natural or even empirical philosophy. All it means is that these rules suffice for reasoning validly from phenomena to inductions in natural philosophy.

Appendix A. Newton’s moon test

Introduction

Given the importance of Newton’s moon test to Book 3, I will step through the test in order to distinguish Newton’s deduction to identical values for lunar centripetal acceleration and terrestrial gravity from his induction to the same cause for these values. Newton’s moon test is provided after prop. 4:

Let us assume a mean distance [of the moon from the earth] of 60 semidiameters in the syzygies; and also let us assume that a revolution of the moon with respect to the fixed stars is completed in 27 days, 7 hours, 43 minutes, as has been established by astronomers; and that the circumference of the [lunar orbit] is 123,249,600 Paris feet, according to the measurements made by the French. If now the moon is imagined to be deprived of all its motion and to be let fall so that it will descend to the earth with all that force urging it by which it is kept in orbit, then in the space of one minute, it will by falling describe 15¹/₁₂ Paris feet. This is determined by a calculation carried out either by using prop. 36 of book 1 or (which comes to the same thing) by using corol. 9 to prop. 4 of book 1. For the versed sine of the arc which the moon would describe in one minute of time by its mean motion at a distance of 60 semidiameters of the earth is roughly 15¹/₁₂ Paris feet … And therefore that force by which the moon is kept in its orbit, in descending from the moon’s orbit to the surface of the earth, comes out equal to the force of gravity here on earth, and so (by rules 1 and 2) is that very force which we generally call gravity. (Newton, 1999, p. 804)

First, we must understand Newton’s terminology, namely, what he means by ‘syzygies’ and ‘versed sine’.

The term ‘syzygy’ is from the Greek ‘suzugia’ which means ‘union’. The lunar syzygy is either of two points on the lunar orbit where the sun, moon, and earth can lie in a straight line. The importance of syzygies lies in the fact that only at the syzygies could astronomers in Newton’s day extrapolate information about the distance between the earth and the moon. Furthermore, if the lunar orbit is approximately circular, then information about the earth–moon distance at a syzygy can be extended to any earth-moon distance on the lunar orbit, making the syzygial earth–moon distance an estimation of the lunar orbit’s semidiameter.

The term ‘versed sine’ is an antiquated mathematical term, best understood with an example. See Fig. 1.

Fig. 1.

Some properties of circles.

Let B represent the centre of a circle, AC an arc, and BC and AB radii. Let Aa ⊥ BC. Then, Ba is the cosine of AC, aC is the versed sine of AC, and if ABC equals θ, θ equals AC / BC. Also, cosθ=Ba / AB.45 These terms are important because Newton intends to show that if the moon travels an arc over some time t while orbiting the earth, which is at the centre of its orbit, then if the moon did not have any inertial velocity, it would travel toward the earth in proportion to the versed sine of that arc over the same time t.

Use Fig. 2 for the deduction to follow.

Fig. 2.

A Representation of lunar orbit.

The deduction

Let C represent the orbit’s circumference, T the moon’s period, r the earth’s radius, v the moon’s velocity, and D the distance between the moon’s and the earth’s centres. Assume the earth and the moon are spheres, the earth’s gravity is concentrated at its centre, the moon’s orbit is uniform and circular, the quotient of D divided by the sum of r plus the moon’s radius is D, and that D is sixty times the magnitude of r.

Since …

C=2πr & r=C/2π

D=60r

v=2πD/T

It follows that …

v=2π (60r)/T=2π(60C)/2πT=60C/T

For Δt=1 unit of time, it follows that …

v∝Δd/1∝Δd∝arc length

Since …

θ=arc length/radius

It follows that …

θ=60C/(T×D)=60C/T(60r)=60C(2π)/T(60C)=2π/T

Since …

cosθ=cosine/radius

It follows that …

cos(2π/T)=cosine/D=cosine/60r=(cosine×2π)/60C

And …

cosine=(60C×cos(2π/T))/2π

Since …

versed sine=radius−cosine

It follows that …

versed sine=D−(60C×cos(2π/T))/2π=60r−(60C×cos(2π/T))/2π=60C/2π−(60C×cos(2π/T))/2π=(60C−(60C×cos(2π/T))/2π=60C(1−cos(2π/T))/2π

Since …

C=123,249,600 Paris feet & T=39,343 …

It follows that …

versed sine=60(123,249,600 Paris feet)(1−cos(2π/(39,343))/2π=15.008 Paris feet

Since the moon’s orbit around the earth is perturbed by an acceleration toward the sun, Newton calculates that any value for the moon’s acceleration toward the earth should be corrected by 1 / 178.725 of that value.46 So, (15.008 Pft×1 / 178.725)+15.008 Pft=15.092 Paris feet.

Since the centripetal force that holds the moon in orbit varies as the inverse of the square-distance, F_c ∝ 1 / r², the centripetal force at earth is to the centripetal force at the moon as 1 / r² is to 1 / (60r)². In that case, the centripetal force at earth is 60² times the centripetal force at the moon. Since force is proportional to change in distance, a change in distance on earth due to the centripetal force is 60² times a change in distance at the moon due to the centripetal force. So, if the moon falls d feet at the moon, it would fall 60² d feet at earth. However, if the moon falls d feet in 1 minute at the moon, that does not mean it would fall 1/60 of 60² d feet in 1 second on earth. Rather, objects fall toward the earth such that the change in distance is proportional to the square of the change in time, Δd ∝ (Δt)². (This is one of Galileo’s discoveries in projectile motion.)47 Now, since ‘Δd ∝ (Δt)²’ on earth, it follows that ‘Δd₁ / (Δt₁)²=Δd₂ / (Δt₂)²’, where subscripts of letters denote intervals. Since the moon would fall 60²×15.008 Paris feet in 60 seconds on earth, the following equation solves the distance the moon would fall in 1 second on earth: (60²×15.092 Pft) / (60 s)²=X Pft / (1s)². Solving for X yields …

X Pft=(602×15.092 Pft)(1s)2/(60 s)2=(602)(15.008 Pft)(12)(s2)/(602)(s2)=15.092 Pft

Hence, the moon would drop 15.092 Paris feet in 1 second on earth.

Christiaan Huygens found that objects on earth drop 15.096±0.01 Paris feet in 1 second. In fact, Huygens discovered that the value for gravitational acceleration is proportional to the distance an object falls in one second divided by half the length of a pendulum beating in Paris, which is 1.52966 Paris feet. So, 15.092 Pft / 1.52966 Pft=9.8662, which is close to Huygens’s value 9.8689±0.01. (Notice that any value ± 0.01 of 9.86 is also ± 0.01 of π².)48

The Induction

Since 9.8662=9.8689±0.01, nature is simple from rule 1, equivalent phenomena are from equivalent causes from rule 2, & 9.8689±0.01 is from earth’s gravity, it follows that 9.8662 is from earth’s gravity.

Conclusion

So, Newton’s moon test deduces that the centripetal acceleration keeping the moon in orbit has the same magnitude as the acceleration of objects toward earth. Newton then makes an induction that those two accelerations are actually the same acceleration.49 Hence, the moon is kept in orbit by earth’s gravity. The moon test is important because it was the first explicit use of the rules of reasoning, it served as the foundation for Newton’s argument for universal gravity, and it served as the foundation for Newton’s argument that the Copernican system of the planets is essentially correct.