Newton's Experimental Proofs as Eliminative Reasoning

ATHANASSIOS RAFTOPOULOS∗

NEWTON’S EXPERIMENTAL PROOFS AS ELIMINATIVEREASONING

ABSTRACT. In this paper I discuss Newton’s first optical paper. My aim is to examinethe type of argument which Newton uses in order to convince his readers of the truth of histheory of colors. My claim is that this argument is an induction by elimination, and thatthe Newtonian method of justification is a kind of “generative justification”, a term due toT. Nickles. To achieve my aim I analyze in some detail the arguments in Newton’s firstoptical paper, relating the paper with Newton’s other writings in optics, and especially hisearly correspondence in defence of his theory of colors.

1. INTRODUCTION

In the two first books of these Opticks, I proceeded by this analysis to discover and provethe original differences of the rays of light in respect to refrangibility, reflexibility, andcolour, . . . . (Newton 1730, Part I, 405)

Newton’s first official announcement of his theory regarding the propertiesof light, however, did not take place in theOpticks, but in a much earlierpaper that was addressed to the Royal Society to be read in one of its reg-ular meetings. This paper is known as Newton’s first optical paper. Thereare several differences between the account of the theory in theOpticksandthat in the first paper, but the one that is of interest for my purposes is thatin the paper Newton provides us with an abundance of details regarding theprocess that led him to believe that he has proved his theory, details thatare absent form theOpticks, given the formal character of the expositionof the theory there. Newton’s paper stirred a lot of heated discussion, andwas widely discussed not only in England but in Europe as well. It gaverise to a substantial amount of correspondence between Newton and otherscientists, including Oldenbereg, Hooke, Collins, Gregory, Moray, Pardies,Linus, Lucas, Flamsteed, and Huygens, a correspondence which lastedfrom early 1672 to 1678. The paper aims at establishing the properties oflight concerning refrangibility and colors, and Newton expounds the wayhe discovered the theory.

One must note from the outset that Newton does not use the term “dis-covery” the way philosophers of science do nowadays, after the radical

Erkenntnis50: 95–125, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

96 ATHANASSIOS RAFTOPOULOS

distinction between the contexts of discovery and proof. Following thetradition of his time, he does not distinguish between “proof” and “dis-covery”. Whenever he refers to his scientific method he starts by statingthat this is a method of discovery and proof. This is exactly how Bacon(1990) and Descartes (Regulae, AT. X; Discourse, AT. VI: Meditationsand Replies, AT. VII) 1 conceived of their method. It is worth mentioningthat this tradition is maintained until Mill, for whom induction becomesthe method of “discovery and proof”.

Newton’s view that discovery and proof go together is not surprisingin the light of his calling his method “analysis” (Newton 1730, 404–405;ADD. 3970.3. Folio 480v, quoted by McGuire 1970, 185). According tothe tradition, analysis ispar excellencethe method of discovery. Newton,however, follows Descartes in breaking the tradition that takes synthesisto supplement analysis, by yielding the proof of what has been discoveredin analysis. As the passage from the Opticks discloses, in Newton’s mindanalysis is probative, that is, yields proof. This means that for Newtonthe same inferences that are employed in order to discover, let us say, aproperty of the rays of light, are the same that establish this property. Inother words, themodeof inference is the same in both cases and there isno distinction between the logic ofdiscoveryand that ofjustification.

Now, let us consider Newton’s own apocryphal account of how the ideaof a universal gravitation was first formed in his mind (the apple, etc.), andhis account of his discovery of the theory of colors in the first optical paper.In the former case, Newton describes how he got for the first time an idea,an idea which he elaborated, refined and tested in order to prove. In thelatter case, Newton refers to the process itself of the elaboration of an idea,and calls this whole process the “discovery” of the different refrangibilitiesof the rays of light. Thus, we have to conclude that Newton’s “logic of dis-covery” should not be taken in the sense of the logic of “theory generation”(Curd 1980, 418–20), that is, of the process by which the scientist hits upona hypothesis, but in the sense of the logic of the process of elaborationof a theory. Newton assumes that this elaboration finally establishes thehypothesis and that the inferences involved, in elaborating the hypothesis,prove it.

If this is what Newton has in mind when he speaks of “discovery”,then he is not using the term in the same way as current philosophy ofscience. Even when philosophers emphasized that the “logic of discovery”should not be taken to cover the process of theory generation, as logi-cal positivism and Popperianism have traditionally thought, they insistedthat it should not be taken to be coextensive with the logic of justifica-tion either. Rather, this logic should be deemed to cover the context of

NEWTON’S EXPERIMENTAL PROOFS AS ELIMINATIVE REASONING 97

prior plausibility or probability of the theory, that is, it should take up theprocess by which a theory is deemed to be worth pursuing, either becauseit is plausible, or highly probable,beforeit is tested by new experimentsspecifically designed to confirm or disconfirm the theory. This is far fromNewton’s idea of discovery; in a draft preface for the 1704Optickshewrites (ADD2.3970.3.Folio 480v, quoted in McGuire 1970, 184–5.):

The method of resolution [analysis] consists in trying experiments and considering all thephenomena of nature relating to the subject at hand and (drawing conclusions from them)and examining the truth of these conclusions by new experiments . . . until you come to thegeneral properties of things.

As this statement shows, experiments testing the propositions drawn fromthe phenomena are an essential part of the method of discovery and proof.

Despite Newton’s claim that his paper lays down the actual way hediscovered the theory of the different refrangibility of colors, this claimdoes not reflect the truth. In discussing the paper we will encounter severaloccasions when Newton’s report of his discovery does not agree with theway he proceeded to elaborate and develop his theory, as the comparisonwith Newton’s own notes in hisTrinity Notebook, in which we can witnessthe formation of his ideas, clearly shows. The way from the experimentsto the theory is not as straightforward as Newton would like us to believe,and the order and the importance of some of the experiments are not asdescribed in the paper.

Furthermore, as Shapiro argues (1993, 85; 172; 175; 176; 178), thevarious causal hypotheses regarding the nature of light, and especiallyNewton’s conception of light as consisting of particles that cause vibra-tions in the medium in which they are propagated (the ether), play anessential role in Newton’s process of discovery of the properties of light.However, whenever Newton set out to present his findings he tried toeliminate all traces of the speculative hypotheses and lay forth only what,he believed, could be proved by experiment, referring to his argumentsas “experimental proofs”. Shapiro distinguishes between the method usedby Newton in reporting and proving his results and the actual methodemployed in the process of discovery of these results. The first opticalpaper portrays the Newtonian method of proving Newton’s theory of theproperties of light.

One could also argue that there is no method of discovery, because theprocesses of elaboration and development of a theory are complex enoughto defy any methodological systematization. The order and the kinds ofinferences involved in this process cannot be put under the auspices of asingle methodology, even with respect to the work of the same scientist. AsSchuster claims (1986, 73–5), a scientist’s methodological account of his


discoveries is but an after-gloss that purports to present a unified picture ofhis thought. Any methodological rules that this scientist claims to consti-tute his method are usually too vague, and at the same time too limited, tobe of any use in developing a theory.

Be that as it may, we can say, following Nickles (Nickles 1988, 35), thatNewton confounds the “historical mode of generation” of a theory withthe “logical generability or derivability” of this theory, which means thatthe order of the arguments and experiments presented in the paper is notthe historical but the logical order which renders the arguments more clear,stronger and less ambiguous. In other words, Newton offers a methodolog-ical reconstruction of his actual research. Thus, Newton offers a linear andretrospective account of his original reasoning, and his “primary concern isto make the evidential argument conform to the methodological canons of aparticular experimental discourse” (Gooding 1990, 6). It follows that New-ton’s main concern is not to present an accurate picture of the formation ofhis theory, but to convince his readers of its truth.

In this paper I will analyze and discuss the experimental method New-ton employs in his first optical paper to convince his audience about thetruth of his theory of white light and colors. I will try to discern the kinds ofinferences involved, and I will claim that the main burden of Newton’s ex-perimental method falls on an eliminative procedure, which allows Newtonto claim that his theory is the only one that fits all relevant phenomena.

In addition, my discussion aims at attacking Achinstein’s (1990) at-tempt to interpret the Newtonian method. Achinstein discusses the in-ferences involved in the “deduction from the phenomena”. Achinstein’sconception of the Newtonian deduction from phenomena is exemplifiedby his reconstruction of Newton’s argument regarding the first propositionin the Opticks: “Lights which differ in colour, differ also in degrees ofrefrangibility” (Newton 1730, 20).

According to Achinstein (1990, 146), the proof of the proposition con-sists of four steps: (a) As a result of experiments the following phenom-enon is established: when the paper, one half of which is red and the otherhalf blue, is viewed through a prism, the blue half is more displaced thanthe red; (b) From axiomVIII of theOpticks(Newton 1730, 18) which statesthat: “An object seen by reflexion or refraction, appears in that place fromwhence the rays after their last reflexion or refraction diverge in falling onthe spectator’s eye.”

From (b) and the discussion following it, we conclude that when anobject is seen through a prism, it is not seen in its proper position but isdisplaced as a result of the refraction of the light in the prism. By ordinarydeduction we infer from (a) and (b) that: (c) the blue rays are more re-


fracted by the prism than the red. Finally, inductively from (c) we infer that(d) blue rays in general are more refrangible than red, and more generallythat the rays of different colors are differently refrangible.

If Achinstein thinks that the above description fits Newton’s methodof justification in optics, then it is doubtful whether it is consistent withthe method Newton actually employed. Achinstein conceives of Newton’sexperimental method as a mixture of enumerative induction, causal simpli-fication, which takes up the inferences discussed by Newton in the first tworules of philosophizing (Achinstein 1990, 141–144), and deduction. I willshow that this description renders little justice to Newton’s experimentaland theoretical ingenuity, in that it omits the most characteristic feature ofNewton’s justifications, to wit, the elimination of alternative theories of therays of light.

I will also claim that even if Achinstein’s reconstruction is meant to berestricted to theOpticks, the account of the method there is still deficient inexactly the same respect, since traces of the argument by elimination canbe traced in theOpticksas well.

As a letter to Cotes (Thayer 1953, 6) reveals, upon the completion of themethod, propositions pertaining to the properties of the things have beendeduced3 from experiments “[t]hese principles are deduced from the phe-nomena and made general by induction”. This assertion can also be foundin a letter to Oldenburgm (Cohen 1958, 93), which reads: “Your know theproper method for inquiring after the properties of things is, to deducethem from experiments”. In that regard, I will examine the notion of the“deduction of the phenomena”, and I will claim that the term ‘deduction’has a much wider meaning that it does nowadays, insofar as it encompassesampliative inferences, such as induction, and analogical inferences.

At this juncture I would like to state what I do not attempt to do in thispaper. First, I am not going to evaluate Newton’s conception of an “exper-imental proof”. Second, as I have already said, I am not going to attackthe issue of Newton’s conception of his method as a method of discoveryand justification at the same time. My paper is restricted to discussing themethod of justification as employed in the first optical paper, and is notconcerned with the way Newton discovered his theory. This paper, finally,addresses only the issue of Newton’s experimental method in optics. Ishall not discuss the Newtonian method employed in Newton’s work indynamics. Therefore, I will not make any claims regarding the relationsbetween the method(s) employed in these two fields of inquiry.


Figure 1. Newton’s initial experiment: a prism is placed near a small hole in the shutters ofthe window of a darkened chamber in such a way that sun light is refracted to the oppositewall. The colored pattern formed on the wall has an oblong form, whereas, according tothe laws of refraction, the pattern should have been circular.

2. NEWTONIAN METHOD AND OPTICS

2.1. The Experiments – The Elimination of Alternative Accounts ofRefrangibility

In his first optical paper (Cohen 1958, 47–78), Newton describes the wayhe discovered and proved that different colors have different degrees ofrefrangibility. The initial experiment, which created the phenomenon to beexplained, consisted in placing a prism near a small hole in the shuttersof the window of a darkened chamber in such a way that sun light wasrefracted to the opposite wall. The experiment resulted in the productionof a colored pattern on the wall having an oblong form, while, according tothe laws of refraction, the pattern should have been circular (see Figure 1).Newton compared the length of the coloured spectrum with its breadth andfound it about five times greater, a result which he called “extravagant”,and which excited him to find its cause.

I would like to note here that what Newton actually saw is open to de-bate. It is certain that the description of the spectrum he relates in his paperdoes not correspond exactly to the actual picture formed on the screen. Itis rather an idealization which “cried out loud for the interpretation thatNewton provided”, as T. Kuhn remarks in his introduction to Newton’soptical papers in Cohen (1958, 34). According to Kuhn, Newton saw afigure much narrower and more pointed at the blue end than at the red.

Before we continue examining Newton’s paper, however, we must di-gress and return to the problem of the discrepancy between Newton’s own


account of the elaboration of his theory of colors, and the actual process ofthe formation of his theory. Newton says in the first paper that the experi-ment with the prism and the resulting elongated spectrum “excited” him tofind the cause of the observed phenomenon. But this is not true. In his earlyoptical notes he reported an experiment in which a woven fabric is paintedhalf blue and the other red. When Newton looked at the object througha prism he noticed that “one half of the thread shall appear higher thanthe other, not both in one direct line”. Newton drew the conclusion thatthis is caused “by reason of the unequal refraction in the differing colours”(McGuire and Tamny 1983, 467–8). The same experiment is repeated inthe Opticks(Newton 1730, 21–3) where it is used in the proof by exper-iment of Proposition I: Theorem I “Lights which differ in Colour, differalso in Degrees of Refrangibility”. Thus, the elongated spectrum did notplay in the discovery the role assigned to it in the paper.

Furthermore, though Newton may have had some reason to believe thatthe spectrum is due to the various angles of refraction of the different color-producing rays, this did not imply anything regarding the heterogeneity ofwhite light. It is still possible that the different colors could be generatedwithin the prism by means of a modification of homogeneous white light,and then be refracted in different angles (as Kuhn reports, this was New-ton’s first idea when he performed the prism experiment; see Cohen 1958,34).4 Indeed, we do not find any mention of the heterogeneity of white lightin these early optical notes. Thus the road to discovery was not as simpleand straightforward as Newton presented it in his first optical paper.

The first set of possible causes that occurred to Newton in the firstoptical paper, though as he says they were not very plausible, was that:

I could scarce think, that the various thickness of the glass, or the termination with shadowor darkness, could have any influence on light to produce such an effect; yet I thoughtit not amiss, first to examine those circumstances, and so tried, what would happen bytransmitting light through parts of the glass of divers thicknesses . . . . (Cohen 1958, 48)

None of these circumstances was found “material”, since the fashion of thecolors was the same in all cases with different thicknesses, a fact whichsuggests that the thickness of the glass is not a causal factor of the phe-nomenon observed.

The second possible cause invoked by Newton is more interesting, sincewhat Newton does is to reject a standard part of any explanation of colorsfrom Anaximenes to Grimaldi, Descartes, Barrows, and Hooke, namelythat colors result from the mixture of light with darkness, a darkness thatsupposedly comes from the boundaries of the hole. To test this suppositionNewton “tried, what would happen by transmitting light . . . through holesin the window of divers bigness, or by setting the prism without so, that


the light might pass through it, and be refracted before it was terminatedby the hole” (Cohen 1958, 48). Again, the fashion of the colors did notchange, a fact that made Newton disregard this supposition. An even morevigorous attack against this “hypothesis of the philosophers” is found inLecture 8of theOptical Lectures, in Shapiro, 1984, 161–3.

The next possible cause Newton considered was that the colors mighthave been thus dilated because of the unevenness in the glass or some otherirregularity. Thinking that if a prism causes an irregular dispersion of thelight rays, a second prism which refracts light in a contrary way shoulddestroy the regular effects of the first prism while augmenting the irregularones, he combined two prisms so that they refracted light in contrary waysand performed an experiment, whose main feature was that the secondprism refracts the light in “contrary ways”, which proved that: “whateverwas the cause of the length, it was not any contingent irregularity“, since“the light which by the first prism was diffused in an oblong form, was bythe second reduced into an orbicular one . . . ” (Cohen1958, 48).

The rejection of this hypothesis carried a special weight for Newton,for it is by means of some kind of irregularities that Descartes and Hookesought to explain the phenomena of colors. According to Hooke, lightis a short vibrating motion in the luminous body. This vibration spreadssymmetrically through the surrounding medium (ether). The pulses are atright angle to the beam of light, unless they find in their way an interfacebounding a different transparent medium. Should this happen, the pulsesare distorted and they cease being normal to the beam. This distortion,Newton’s irregularity, of the pulses constitutes color, blue being the resultof an oblique and confused pulse whose weakest part proceeds, and red be-ing the result of a distorted pulse whose strongest part proceeds. Newton’smention to Descartes refers to a non-standard explanation of the color ofthe tail of comets, put forth in the third part ofThe Principles(AT. IX:185–88).

Newton’s argument, however, is unfair to Hooke. In Hooke’s theory,refractions depend on the geometry of the refracting body. Thus, it is im-possible for Hooke to have held that a second prism, placed in the oppositesense to the first prism, should augment the effect of the first refraction.Indeed, in hisMicrographiaHooke stated that a contrary refraction wouldrestore the pulse to its prior simple and uniform form, which is exactlythe result of Newton’s experiment. Thus, the argument does not carry anyweight against Hooke. For such an argument we have to look at anotherexperiment that Newton reports in his answer to Hooke’s objections toNewton’s first optical paper (see Figure 2). There (Cohen 1958, 125) New-ton describes an experiment with two crossed prisms. If Hooke’s theory


Figure 2. The crossed experiment that shows the falsity of Hooke’s theory: The length axisof both prisms are mutually perpendicular. The spectrum reappears on the wall, being onlydisplaced and turned a certain angle (spectrum B), with respect to the spectrum that wouldhave resulted without the second prism (spectrum A). If Hooke was right, the refractedbeam should be stretched transversely with respect to the beam refracted by the first prismalone.

of colors was correct, the refracted beam should be stretched transverselywith respect to the beam refracted by the first prism alone. Newton ob-served that the spectrum of the second beam is the same as that producedby the first prism alone, the only difference being that the second spectrumis inclined at an angle of 45◦ to the axis of the second prism. Thus:

And by these observations, since the breadth of the image was not augmented by thecross refraction of the second prism, that refraction must have been performed withoutany splitting or dilating of the ray; and therefore at least the light incident on that prismmust be granted an aggregate of rays unequally refrangible . . . .5

That Newton has in mind, when he deals with this issue, Descartes andHooke’s hypotheses, is shown in Newton’s answer to Hooke’s criticismof his first optical paper. This experiment, Newton says, was designed toshow

that the length of the coloured image proceeded not from any unevenness in the glass, orany other contingent irregularity in the refractions. Among other irregularities I know not,what is more obvious to suspect, than a fortuitous dilating and spreading of light after suchmanner, as Descartes has described in his ethereal refractions for explicating the tail ofcomets; or as the Animadversor [Hooke] now supposes to be effected by the splitting and


rarefying of his aethereal pulses. And to prevent the suspicion of any such irregularities,I told you that [a description of his experiment in the first optical paper follows]. (Cohen1958, 124)

Considering another possible cause, Newton writes:

Then I began to suspect, whether the rays, after their trajection through the prism, did notmove in curved lines, and according to their more or less curvity tend to divers parts of thewall. (Cohen 1958, 50)

Newton gives the reason that made him think that the above might be a pos-sible explanation of the phenomenon at issue; this was that he rememberedthat he had seen a tennis ball, struck with an oblique racket, describing sucha curved line. As he writes:

For, a circular as well as a progressive motion being communicated to it by that stroak,its parts on that side, where the motion conspires, must press and beat the contiguousair more violently than on the other, and there excite a reluctancy and reaction of the airproportionally greater. And for the same reason, if the rays of light should possibly beglobular bodies, and by their oblique passage out of one medium into another acquire acirculating motion, they ought to feel the greater resistance from the ambient aether, onthat side, where the motions conspire, and then be continually bowed to the other. (Cohen1958, 50)

But,

notwithstanding this plausible ground of suspicion, when I came to examine it, I couldobserve no such curvity in them.

Again Newton’s discussion is a cautious attack against an existing hy-pothesis purporting to explain the phenomena of light rays. This is thehypothesis put forth by Descartes in hisOpticks. There, (Descartes, AT.VI: 88–9; CSM. 1:155)6 in order to explain refraction, reflection, and col-ors, Descartes appeals to supposed changes in the speed of the small ballsthat constitute light, when they pass from one medium to another with adifferent density. Newton attacks the account of colors Descartes gave inthe Meteorologywhere color was associated with the various rotationalspeeds that the particles of light acquire when they pass from one mediumto another (Descartes 1966, 189–93).

Newton was, of course, aware that the phenomenon of the elongationof the spectrum of colors had been observed before. However this elon-gation was seen in a different light. First, the observed elongation wasmuch smaller than the one reported by Newton. This was due to the factthat the screen was placed close to the prism and this resulted in a smallerelongation. Second, an explanation of this elongation had been put forwardby standard theories of colors (according to which colors result from themodification of light when it interacts with bodies). This elongation, it was


claimed, was due to the fact that the sun has finite dimensions, so that thelight rays falling on the prism were not parallel, but instead, had differentangles of incidence. Hooke, for instance, believed that the divergence ofthe rays is caused by the fact that the sun is not a point source of light, andthat the resulting divergence could be accounted for by his theory.

Newton, in order to claim that this phenomenon is the litmus test thatdistinguishes his explanation of the dispersion of colors from other in-correct explanations, had to show that his opponent’s explanation of thephenomenon was not satisfactory. Thus, he modified the configuration ofthe experiment so that the rays coming from opposite parts of the sun’sdiscus were virtually parallel to each other. He found out that the differ-ence between the two configurations could account for 31 or 32 minutes ofdivergence (which is the angular size of the sun), and thus the convergenceof the beam incident on a small hole. This is much less than the observedelongation of the spectrum, 2 degrees and 49 minutes. Though this cal-culation revealed the insufficiency of the other theories of color, Newtonproceeded to perform another experiment:

and having placed it [the prism] at my window, as before, I observed, that by turning it alittle about its axis to and fro, so as to vary its obliquity to the light, more than an angle of4 or 5 degrees, the colours were not sensibly translated from their place on the wall, andconsequently by the variation of the incidence, the quantity of refraction was not sensiblyvaried. (Cohen 1958, 49)

This experiment, Newton concludes, makes evident that:

the difference of the incidence of rays, flowing from divers parts of the sun, could not makethem after decussation diverge at a sensibly greater angle, than that at which they beforeconverged; which being, at most, but about 31 or 32 minutes, there still remained someother cause to be found out, from whence it could be 2 degr. 49′. (Cohen 1958, 49–50)

2.2. Methodological Considerations – Newton’s ‘Positivism’

The last step in Newton’s experimental proof is theexperimentum crucis.Before we examine it in detail, we can note that our discussion thus farprovides some information about the experimental method that Newtonbelieves will persuade his reader: facing a phenomenon in need of ex-planation, Newton derives various possible causes that could explain thephenomenon. Then he tests them against other relevant phenomena. New-ton does not claim that the initial inference to a cause suffices to establishthe proposed cause; instead, further testing is required to eventually es-tablish the cause. In this sense the possible causes that Newton initiallyderives are tentative conclusions. The causes that fail to account for all thephenomena are one by one eliminated through controlled experimentation.


We can see that Newton’s method in the first optical paper conforms withhis general methodological account both in the Opticks (Newton 1952,404–405):

analysis consists in making experiments and observations and drawing general conclusionsfrom them by induction, and admitting of no objections against the conclusions, but suchas are taken from experiments, or other certain truths.

and in a draft of an intended preface for the 1704 Opticks (ADD. 3970.3.Folio 480v quoted by McGuire, McGuire 1970, 185):

The method of resolution consists in trying experiments and considering all the phenomenaof nature relating to the subject at hand (and drawing conclusions from them) and exam-ining the truth of those conclusions by new experiments and drawing new conclusions(if it may be) from those experiments and so proceeding alternately from experiments toconclusions and from conclusions to experiments until you come to the general propertiesof the things.

A typical example of such an inference can be found in Newton’s dis-cussion of the curvature of the trajectory of the light rays through the prismas a possible cause of the phenomenon of the spectrum of colors. Newtongives an argument for the plausibility of such a cause that is a clear case ofthe application of causal simplification (Cohen 1958, 48):

And it increased my suspicion, when I remembered that I have often seen a tennis ball,struck with an oblique racket describe such a curve line . . . And for the same reason, if therays of light could possibly be globular bodies . . .

Newton’s inferences to the possible causes of a phenomenon yield variousdegrees of plausibility. When Newton comments on the various thick-nesses of the glass of the prism, he begins by writing that he “could scarcethink that . . . ” (Cohen 1958, 48). In contradistinction, when it comes toanother possible cause, namely the curved trajectory of the light particles,Newton writes that “it increased my suspicion . . . ” (Cohen 1958, 48).

We should be careful at this point not to identify a possible cause, asconceived by Newton, with a mere hypothesis that is to be tested againstexperience. According to the precepts of the Newtonian methodology, thepossible causes are the result of inductive inferences from the phenomena.Unlike hypotheses, the only support of which comes from the fact thatthey could explain the phenomena, Newton believes that these conclusionshave, in addition, a kind of direct support by virtue of their being inferencesfrom phenomena, inferences that are none other than the ones described inthe first three rules of reasoning. This trait sets Newton’s method apartfrom the method of hypothesis.

Furthermore, Newton is not willing to admit among the possible causesof the phenomena causal hypotheses regarding the nature of light and col-


ors, at least at the first phase of the inquiry. As he writes to Pardies (Cohen,1958, 106):

[I]t is to be observed that the doctrine which I explained concerning refraction and colours,consists only in certain properties of light, without regarding any hypotheses, by whichthose properties might be explained. For the best and safest way of philosophizing seems tobe, first to inquire diligently into the properties of things, and establishing those propertiesby experiments and then to proceed more slowly to hypotheses for the explanation of them.For hypotheses should be subservient only in explaining the properties of things, but notassumed in determining them; unless so far as they may furnish experiments.

Let me elaborate on Newton’s conception of causes, as the nature ofthings, and of causes, as the properties of things. In the Opticks he distin-guishes between levels of causes:

By this way of analysis we may proceed from compounds to ingredients, and from mo-tions to the forces producing them; and in general, from effects to their causes, and fromparticular causes to more general ones, till the argument end in the more general. This isthe method of analysis. (Newton 1952, 404–405)

The properties7 of things are at the first level of causes, in that theyaccount for the observed behavior of things. To the question “why do theplanets describe ellipses around the sun?” Newton’s answer is “becausethey follow the inverse square law of gravity” However, the answer tothe question “why does gravity act according to this law?” is unknownto Newton, since thecause(cause at the highest level), of gravity has notbeen discovered. We can say following Newton that at the highest level ofcauses we have accounts of the nature of things.

Gravitation towards the sun is made up out of the gravitations towards the several particlesof which the body of the sun is composed; and in receding from the sun decreases accu-rately as the inverse square of the distances . . . But hitherto I have not been able to discoverthe cause of those properties of gravity from phenomena, and I frame no hypotheses.(Newton, 1729, 547)

The reader may have noted that Newton’s distinction betweenproper-ties and causes corresponds roughly to the distinction between phenom-enological and causal accounts, as is used in modern scientific practice.According to it, a law is observational or phenomenological, if it is used todescribe rather than explain the behavior of a system. A law which purportsto explain is called theoretical or fundamental (Cartwright, 1983). Theinverse square law, is aproperty of things, as are the laws of refraction,reflection, and the prismatic analysis of the light. The nature of gravity(that is, the forces that cause it), as well as the nature of light and of colors,are thecausesof the corresponding phenomena. Thus, when I say thatNewton rejectscausalhypotheses, the term “cause” should be understood

https://www.researchgate.net/publication/283420415_How_the_Laws_of_Physics_Lie?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==


as signifying the “nature” of things (causes at the highest level) and not thepropertiesof things (causes at level one).

The same claim can be made with regard to Newton’s theory of opticalphenomena. In the same letter to Pardies Newton writes “Hence it has beenhere thought necessary to lay aside all [causal] hypotheses”, and in theOpticks, writes that he proves the properties of light “without determiningwhat light is, or by what kind of force it is refracted” (Newton 1730, 81–2). Newton wants to avoid any uncertain causal explanation of the opticalpropertiesof the light rays, and try to establish these properties by meansof experiments, “to deduce them from experiments”. Theseproperties,furthermore, must be established independently of anycausalpresuppo-sitions, and this is why Newton insists in his letter to Pardies that he laysaside allcausalspeculations.

Newton confirms his firm distinction between thecausesof the phe-nomena and thepropertiesof these phenomena. Thepropertiesof lightinclude the laws of reflexion and refraction, and the analysis of whitelight, and thecausal hypotheses, or explanations, consist of accounts of thenature of light, whether, for instance, the light consists of waves or smallparticles, and the nature of colors. As far as thecausalhypotheses areconcerned, not only does Newton not make any claims, but he repeatedlyrejects Pardies’ and Hooke’s claims, among others, that his [Newton’s]theory presupposes the corpuscular nature of light. Only in hisQueries, didNewton indulge in propounding his own viewpoints regarding the natureof light and colors, and even in that case he was very careful to point outimmediately that these ideas are speculations for further research.

Shapiro places much weight upon this trait of Newtonian method; hedoes not hesitate to call this positivistic attitude one of the main character-istics of the Newtonian method (Shapiro 1993, 85):

[h]aving deduced the properties [of the light rays], he eliminates all traces of the hypothesis[the causal account of these properties] . . . This was a consequence of his methodology:his refusal to mix principles and hypotheses and the consequent requirement that propertiesbe “abstracted” from hypotheses.

and again (Shapiro 1993, 137; see also 173; 176; 178; 195),

[w]ith the theory of fits, Newton had succeeded in carrying out the dictates of his method-ology by distinguishing the properties of periodicity from the hypothetical nature of thevibrating medium.

This “positivistic” attitude has an important consequence for Newton’soptics, as Newton is not bound to offer acausal theory that unifies andexplains all the known optical phenomena. At the same time, it allowsNewton to disregard any objections to his theory that do not come from


experimental observations.8 When Hooke, for instance, suggests his the-ory, based on the wave interpretation of light, Newton considers it only torefute experimentally its consequences regarding the properties of the raysof light, and refuses to discuss whether light wave- or particle-like.

It should be noted here that Newton’s “positivism” is different frompositivism. The latter largely regarded questions concerning reality as irrel-evant to science, while Newton was profoundly committed to unravellingthecausal mechanismsof nature, as theQueriesin the end of theOpticks,and his papers reveal. He was only careful not to blend unproved specula-tions concerningcausalmechanisms with thepropertiesthat he believedwere proved. Thecausal hypotheses, far from being a hindrance to science,were part of a research program that could eventually establish them asprinciples (Newton 1730, 405). McMullin (1978, 67–74) and Achinstein(1990, 149–51) reach the same conclusion regarding Newtonian “positivis-m”, and the role of hypotheses within the Newtonian system.

If thesecausalhypotheses constitute, for Newton, a legitimate part ofthe scientific research program, as the letter to Cotes suggests, then theymust be amenable to experimental proof, which means that they must bediscovered according to the precepts of the method; that Newton believedsuch proofs possible, is made clear in theQueries. When Newton describeshis method of analysis, he points out that this method can be used not onlyto derive theproperties, of things, but the more general causes (thecausalhypotheses) of these properties as well: “By this way of analysis we mayprocee . . . from effects to their causes, and from particular causes to moregeneral ones, till the argument end in the more general” (Newton 1952,404–405).

Thus, Newton does not reject thecausalhypotheses because they can-not be tested experimentally, in fact he does exactly that by showing ex-perimentally that Descartes’ and Hooke’scausaloptical explanations arewrong, but because he wants to establish first the properties of light inde-pendent of any causal assumptions.

Newton’s firm distinction between thepropertiesof light that can beproved from experiments, and the uncertain hypotheses regarding the na-ture of light, led him to disregard any possiblecausalexplanations of theproperties of the rays of light. Now, Descartes’ and Hooke’s theories oflight figure prominently in the list of thecausalhypotheses of light, sincethey explain the phenomena of light and colors in terms of the movementof small globules, and in terms of vibratory motions respectively. Thesetheories are the main opponents to his theory and Newton knows well thatit would not be convincing to reject them only on thea priori grounds ofhis distinction betweenpropertiesandcausalhypotheses. As we said, even


causal hypotheses can be put to experimental test and Newton does exactlythis in the first optical paper. Thus, Newton tests Descartes’ and Hooke’scausal hypotheses of light that had been proposed.

Let us return to the tentative conclusions mentioned above. These arethe conclusions of weak inferences that do not establish their results withcertainty, as opposed to the strong inferences that do yield the maximumcertainty that is possible in experimental science. We have said that thepossible causes of phenomena are inferred from them by means of ana-logical inferences. These inferences allow us to assume that a cause whichis known to operate in a certain case also operates in a similar case. Thesuccess of an analogical argument hinges heavily upon the extent to whichthe “target” is similar enough to the “base” to justify the extension ofthe cause to the new domain. Thus, analogical arguments alone do notyield any significant certainty. In such a situation we have some reasonto believe thatX is the cause of the examined phenomenon. At the sametime, we may have reason to believe that a different and incompatible causeoperates in our case. The conclusions of weak inferences have the form “x

may be the cause ofF ”, as opposed to strong inferences in which we havestrong reasons for believing that “x is the cause ofE”. A series of weakinferences, finally, cedes its place to a strong inference by means of a seriesof experiments, which seek to test the truth of these inferences, and whichculminate in theexperimentum crucis.

2.3. The Experimentum Crucis – Newton’s Assumptions

It is to Newton’s correspondence with Pardies and Lucas that I think weshould turn now to understand the role of these experiments in the New-tonian method. There, he gives a lucid account of the role of experiments,and of the way they function. On 21 May 1672, Pardies sent his secondletter to Newton, raising the following objection:

But since I now see that it was in that case that the greater breadth of the colours wasobserved, on that head I find no further difficulty. I say on that head; for the greater lengthof the image may be otherwise accounted for, than by the different refrangibility of therays. For according to that hypothesis, which is explained at large by Grimaldi, and inwhich it is supposed that light is a certain substance very rapidly moved, there may takeplace some diffusion of the rays of light after their passage and decussation in the hole.(Cohen 1958, 104)

Newton’s answer came in the same year Cohen 1958, 106–107, empha-sis added):

Hence it has been here thought necessary to lay aside all hypotheses . . . that the force ofthe objection should be abstractly considered, and receive a more full and general answer.By light therefore I understand, any being or power of being . . . which proceeding directly


from a lucid body, is apt to excite vision. And by the rays of light I understand its leastor indefinitely small parts, which are independent of each other . . .This being premised,the whole force of the objection will lie in this, that colours may be lengthened out bysome certain diffusion of light beyond the hole, which does not arise from the unequalrefraction of the different rays, or of the independent parts of light. And that the image isno otherwise lengthened, was shown in my letter in Numb. 80 of the Transactions; and toconfirm the whole in the strictest manner, I added that experiment now known by the nameExperimentum Crucis.

Leaving aside the issue about hypotheses, let us concentrate on New-ton’s response to Pardies’ objection. Let us try to reconstruct Newton’sargument. Facing an objection as to the cause of refraction, he repeats theway he had used to prove, in his first optical paper, that the rays of differentcolors have different degrees of refrangibility. This proof consists of twoparts: (a) a set of “premises” concerning what Newton understands by“rays of light”, and (b) a series of experiments and theexperimentum cru-cis.Both parts play the same role, namely to eliminate possible alternativeexplanations of the phenomena of refrangibilty.

The premises to which Newton refers in his second letter to Pardies arean essential part of Newton’s experimental proofs. They include the factthat the rays of light travel in straight lines, that they consist of small parts,and that they are independent of each other. Another assumption usedby Newton is found in Book One, Part f, of the Opticks (Newton 1730,75). After experiment 15, and the discussion in which Newton provesproposition VI, he concludes (Newton 1730, 81-2):

And this demonstration being general, without determining what light is, or by what kindof force it is refracted, or assuming any thing farther than that the refracting body acts uponthe rays in lines perpendicular to its surface; I take it to be a very convincing argument ofthe full truth of this proposition.

Even if Newton was right to think that the premises presupposed by hisproofs did not commit him to any particular causal hypothesis about light, aproblem still remains. These premises are mere suppositions. Newton doesalmost nothing to prove them. In his letter to Pardies he says that, sincesome parts of light may be intercepted and refracted or reflected withoutthe others, he assumes that the least parts of the rays of light must beindependent. No justification is offered for the other supposition, namelythat light travels in straight lines. Apparently this is because Newton heldthat none was required, since this property of light was well entrenchedin the background knowledge. The premise that the refracting body actsupon the rays in lines perpendicular to its surface is not easily justifiable,and Newton does not offer any justification. This premise much facilitatesthe mathematical calculations (Newton had studied the motions of bodiesunder these conditions inBook One, section 14 of thePrincipia), but this is


hardly a justification of the assumption. Newton knows, however, that noone would reject his proof by denying this assumption, precisely becauseit figures in his opponents’ theories. Descartes’ analysis of the motion oflight rays, based on his ball model in theOpticks, by means of which hederived the sine law, relies heavily on this assumption (AT. VI: 93–108;CSM. 1:156–164).

This last remark brings us to another characteristic of Newton’s as-sumptions upon which his proofs are premised. Newton knows that theseassumptions are commonly held by all involved in the study of opticalphenomena, so it is unlikely that his demonstrations would be contested onthese grounds. Some of the premises are so well entrenched in the back-ground knowledge that Newton feels no need to justify them. For someothers, he offers a justification that consists in a kind of rationalization ofour experiences. For instance, we see that some part of the light may berefracted or reflected when the rest is not; thus, we may conclude that lightconsists of independent parts. The conclusion is so obvious in view of ourexperience, that Newton need not add anything else to justify it. Even thepremises that cannot be grounded directly on experience, and which aregiven no justification, are accepted by Newton’s opponents, and thus donot risk rejection.

These characteristics of the premises at issue remind us vaguely of thefeatures of Newton’s phenomena. Though there is no published definitionof the term ’phenomena’, Newton laid down such a definition intended forthe second edition of the Principia.

Phenomena I call whatever can be seen and are perceptible whatever things can be per-ceived, either things external which become known by the five senses, or things internalwhich we contemplate in our minds by thinking. A fire is hot . . . I am and I think. (McGuire1966, 238–9)

We now see the similarities between Newton’s phenomena and his pre-mises. Most of his premises may be considered to be the result of unprob-lematic inferences from experience. Moreover, they all are indisputable byscientists. In that sense Newton may be justified, in his own eyes, in believ-ing that these premises are not hypotheses, since for Newton, a hypothesisis something that is not among the phenomena, or has not been deducedfrom them. Hence, Newton may claim that his demonstrations are not un-dermined by the role these premises play in them. It goes without saying,of course, that the consensus of the scientific community hardly constitutessufficient reason for deeming these presuppositions unproblematic withoutfurther evidence for them, and in that sense Newton’sdeductionsfrom thephenomena are not probative.


Let us return to the letter to Pardies. The ‘premise’, Newton clearlythinks, blocks the whole class of alternative explanations that could havearisen from different accounts of the rays of light. This being done, the onlypossible alternative remaining is (or, as Newton put it “this being premised,the whole force of the objection will lie in this”) that “colours may belengthened out by some certain diffusion of light beyond the hole, whichdoes not arise from the unequal refraction of the different rays”. WhatNewton has in mind are the explanations of colors by Grimaldi, Descartes,and Hooke, which were the main hypotheses to account for the phenomenaof colors at that time.

The series of experiments reported in the first optical paper, however,show that such a diffusion cannot account for the phenomenon (Cohen1958, 49). Thus, Newton can claim he has shown that the phenomenoncannot be accounted for by these alternatives. Now it is the turn of theexperimentum crucisto play its part. This experiment confirms, in thestrictest manner, that the lengthening of the image is due exclusively tothe different degrees of refrangibility of the rays of light.

The Experimentum Crucisis the last step in Newton’s proof that thecause of the image could only be that light consists of rays that are dif-ferently refrangible (Cohen 1958, 50–51). The gradual removal of the pre-vious possibilities, says Newton, led him to theexperimentum crucis(seeFigure 3):

I took two boards, and placed one of them close behind the prism at the window, so thatthe lights might pass through a small hole, made in it for the purpose, and fall on the otherboard, which I placed at about 12 feet distance, having first made a small hole in it also,for some of that incident light to pass through. Then I placed another prism behind thissecond board, so that the light, trajected through both the boards, might pass through thatalso, and be again refracted before it arrived at the wall. This done, I took the first prismin my hand, and turned it to and fro slowly about its axis, so much as to make the severalparts of the image, cast on the second board, successively pass through the hole in it, thatI might observe to what places on the wall the second prism would refract them.

After noticing that the light tending to the one end of the image after thefirst refraction underwent in the second prism a refraction considerablygreater than the light tending to the other end, Newton concludes that:

And so the true cause of the length of that image was detected to be no other, than thatlight consists of rays differently refrangible, which, without any respect to a differencein their incidence, were, according to their degrees of refrangibility, transmitted towardsdivers parts of the wall. (Cohen 1958, 125)

Let us look at this experiment. Its configuration consists of two paral-lel prisms through which the light passes successively. The first prism isslowly turned about its axis and the resulting image of the light is observed


Figure 3. The Experimentum Crucis: Parallel white light enters a system of two successiveand mutually perpendicular prisms. The light tending to the one end of the image after thefirst refraction undergoes in the second prism a refraction considerably greater than thelight tending to the other end.

on the second board. If Newton’s theory was correct, then the second prismshould augment the result of the first prism, in the sense that the rays ofwhite light refracted in different degrees and analyzed by the first prismundergo a second refraction which, because of the variation of the degreesof refrangibility, causes an even greater spreading of the beam. The unevenrefraction of the rays is observed, confirming that the light consists of raysdifferently refrangible.9 The experiments before theexperimentum crucisproved the falsity of the main rival theories, but did not confirm New-ton’s theory. Theexperimentum crucisaccomplishes that, by confirmingthe uneven refraction of the light rays. Since these rival theories were theonly alternative ones, given the initial assumptions regarding the light rays,Newton can now claim that his experiments show the falsity of all the othertheories, while confirming his. Thus, he claims that his experiments provehis theory. This is why Newton writes to Pardies that the crucial experimentconfirmsin a strict manner the theory.

2.4. Further Methodological Considerations

Further information, regarding the role of crucial experiments in experi-mental proofs, comes from Newton’s answer to a criticism of his refractionexperiments by Lucas (Lucas to Oldenburg, May 1676, in Turnbull, Vol. 2,10; Cohen 1958, 163–168). At the same time this correspondence provides


us with the opportunity to discuss further the Newtonian methodology. Inhis letter Lucas describes a list of experiments in which the results ex-pected by Newton’s account of light did not occur. Newton’s answer (New-ton to Oldenburg, August 1676, in Turnbull, Vol. 2, 79–80; Cohen 1958,173–174) clarifies his notion of the crucial experiment, and its importance.

But yet it will conduce to his more speedy and full satisfaction if he a little change themethod which he has propounded, and instead of a multitude of things try only theExper-imentum Crucis[Newton’s emphasis]. For it is not the number of experiments, but weightto be regarded; and where one will do, what need many?

At first sight Newton’s answer is, at least, incomprehensible. After all,Newton himself thinks, and states it explicitly in both thePrincipia10 andtheOpticks(“admitting of no objections against the conclusions, but suchas are taken from experiments”) that the only objections to his theory maycome only from experiments that provide negative results with respect tohis theory. Lucas does exactly this. In his first paper (Lucas to Olden-burg, May 1676, in Turnbull, Vol. 2, 10; Cohen 1958, 163–168) reportsa series of experiments that challenge Newton’s thesis that different color-producing rays refract differently. He performed experiments and foundthat the elongation of the spectrum is not five times greater than expected,as Newton reports, but only three and a half (Lucas to Newton, March1677, in Turnbull, Vol. 2, 78–80). He observed that there can be refractionwithout colors (Lucas to Oldenburg, October 1676, in Turnbull, Vol. 2,107). He also pointed out, as I have already said, that even if Newton’scrucial experiment yields the results reported by Newton, this only meansthat the rays are refracted unequally (Lucas to Oldenburg, October 1676, inTurnbull, Vol. 2, 104). The experiment does not prove anything regardingthe diverse refrangibility of the rays themselves. Thus, not only had Lucasperformed experiments that seem to disconfirm Newton’s results, but hehad made some measurements too that went against Newton’s own.

Newton’s answer to Lucas’ first letter therefore, shows Newton violat-ing his own methodological percepts, by refusing to consider recalcitrantevidence. But, as Westfall warns us (Westfall 1966), things are a bit morecomplicated. According to Westfall, Newton’s answer must be read withinthe context of the acrimonious correspondence between Newton and Linusand his successors in the University of Liège, and the way Newton dis-misses Lucas objections should be attributed rather to his bitterness againstthe followers of Linus rather than to a violation of his methodologicalpercepts.

In other letters, a ‘more calm’ Newton tries to show that Lucas’ exper-iments are inconclusive or faulty. In his letter to Oldenburg of November1676 (in Turnbull, Vol. 2, 185), for instance, Newton attempts to show why


the elongation of the spectrum observed by Lucas was only three and a halftimes longer than that expected. He does that by stating the various reasonsthat may have made Lucas’ experiment go astray (the angle of the prism,the concavity of the sides etc.)11 Newton is trying to refute Lucas’ resultsand defend his own theory. At the same time, however, we must observethat Newton does not meet all of Lucas’ challenges (he does not answer,for example, Lucas’ objection that there can be refraction without colors).Those issues he passes in silence. Thus, if Newton observed his own per-cept, he should either try to refute all of Lucas’ counterevidence, or heshould, at least, state his theory noting the exceptions. The fact that he didneither of these two things shows that he violated his own methodologicalpercepts.

Let us return now to theexperimentum crucis.It is an experiment thatallows us to drawpositivelya conclusion. It is “positive” exactly becauseit shows that Newton’s theory withstands the tests all of its rival alterna-tives have failed. Newton also subjected his theory to tests other than theexperimentumm crucis to see whether it withstands other phenomena aswell. These are not reported in the first optical paper, but can be found inhis Questiones Quaedam Philosophicaeand his early essay On Colours(McGuire and Tamny, 1983, 329–465 and 466–89 respectively).

We have said in the introduction that Newton’s paper exemplifies theprocess of logical reconstruction of the actual experimental work. In thatrespect one might wonder whether theexperimentum crucisis just a partof such a reconstruction, whether, that is, the relevant experiment wasconceived from the beginning as a logical crucial moment, or whether it re-ceived the status of ‘crucial’ at the stage of the later logical reconstruction.The textual evidence is unclear at this point. As I just said, Newton put histheory to other tests as well, and there is no way of knowing whether theexperimentum cruciswas deemed to have a special status. Furthermore,as the description of the paper makes clear, the experiment is qualitativerather than quantitative. Newton did not make any measurements to showthat a ray is refracted at the second prism exactly at the same degree as atthe first, a fact which is sharply contrasted with Newton’s careful measuresin order to show that the elongation of the spectrum cannot be attributedto the angular size of the sun. Reading all these together with the remarkswe made in the beginning of the second part of this paper regarding theimportance of the elongated spectrum for Newton’s theory of white light,I think that we see the crucial experiment as a part of Newton’s logicalreconstruction of his research in optics.

With this crucial experiment Newton concludes the proof of his theoryconcerning the different degrees of refrangibilities of the different col-


ors, and the heterogeneous nature of white light. This proof is consideredby Newton to be adeductionfrom the phenomena, in the sense that theproperties of light are conclusively derived from the phenomena.

Newton’s proof, of course, relied on the particular set-up he used inelaborating his theory. This proof can be inductively generalized anddeemed to be a general proof of the properties of light Newton discussed.Hence, Newton’s general theory is first deduced from the phenomena, bymeans of the experimental proof, and then is inductively generalized. Thus,as Newton put it, his theory is a “deductionmade general byinduction”.The demand, finally, that a possible cause should be deemed the true cause,if and only if it is the only one that explains the phenomena, distinguishes,among other things, the Newtonian method from the method of hypothesis.

I used before the term ’conclusively’ to describe the kind of the deriva-tion from the phenomena that characterizes Newton’sdeductions.This“conclusively” needs to be qualified. Newton is well aware that experi-mental proofs do not yield the absolute certainty of mathematical demon-strations. They do yield, nevertheless, the maximum certainty that can beachieved in natural science.

And although the arguing from experiments and observations byinductionbe no demon-stration of general conclusions; yet it is the best way of arguing which the nature ofthings admits of, and may be looked upon as so much the stronger, by how much theinduction is more general. And if no exception occurs from phenomena, the conclusionmay be pronounced generally. But if at any time afterwards any exception shall occur fromexperiments, it may begin to be pronounced with such exceptions as occur. (Newton 1730,404–405)

Newton is willing to accept the possibility of some negative evidenceagainst his theory of light. Though he does not state the reasons of sucha possibility, it seems that they must be related with the inherent uncer-tainty involved with experimentation, the inductive character of the infer-ences from the observations, and with the role of the initial assumptionswhich shape the search space for possible causes. Although Newton seemsto be certain about the truth of these assumptions, his personal feelingshardly qualify as legitimate logical grounds for the certainty invested uponthem. Furthermore, as I have mentioned, the fact that these assumptionswere not likely candidates of criticism, does not make Newton’s argumentany stronger. Despite the rhetoric motivating the Newton’s presentation ofthese assumptions, he must have been aware of the problems they posefor his argument so long as they remain unproved. All these may result inan incomplete enumeration of the alternative causes, undermining thus thestrength of the argument.

These considerations lead to the following problem: Newton considerssuch experimental techniques or arguments to be probative, so long as this


is possible in natural philosophy. This is the case only under the provisothat all the possible alternatives have been eliminated. How could Newtonbe certain about that? I shall not fully discuss this problem here, restrictingmyself to three short comments.

First, Newton does not expect his experiments to eliminate allpossiblecauses. These experiments are meant to test all those alternatives that arecompatible with certain premises presupposed in the context of inquiry.The different degrees of refraction of the various colors may be causedonly by the different refrangibilities of the colors or by certain diffusionsof light, given the premise regarding the constitution of the rays of light,namely, that the rays of light consist of indefinitely small parts, which areindependent of each other. It is clear in the letter to Pardies that Newtondoes not have the slightest doubt that his theory about the cause of thedispersion of colors is true. It is, therefore, evident that, in addition tothe aforementioned controlled experiments, his experimental method reliesheavily upon certain assumptions that Newton deems as unproblematic.

Second, there is another characteristic of Newton’s method that restrictsfurther the problem space of possible explanations. This is no other thanNewton’s famed positivistic attitude towards causal explanations (Hall1993, 59), which led him to distinguish the properties of things from un-certain speculations regarding their causes, and which we have alreadydiscussed. Newton is restricted to searching only for the properties of therays of light. This reduces significantly the problem space, and makes thedemand of an exhaustive enumeration of alternative explanations moreplausible.

A third factor restricting the search space is Newton’s conception ofthe analogy of nature, as expressed by the two first rules of philosophizing(Newton 1729, 398–400). Since the possible alternatives must be inferredfrom the phenomena, Newton has to consider only “properties” (and notcausal hypotheses) that are known to work in similar cases.

2.5. The Argument by Elimination and the Opticks

The eliminative argument can be found in theOpticks, which is writtenin a geometrical format according to the standards of the era. Though theeliminative procedure in theOpticksis not as clear as in the optical papers,it is still easily discernible. The way this elimination is managed by meansof a series of experiments is disclosed in the two first books of theOpticks.Let us discuss the proof ofproposition I, theorem Iin the first book oftheOpticks(Newton 1730, 18–26). This proposition reads: ‘Lights whichdiffer in Colour, differ also in Degrees of Refrangibility’ (Newton 1730,20). In the next line Newton writes: “The PROOF by Experiments” and


then proceeds by describing two experiments which, given Newton’s aboveassertion, were deemed to provide conclusive evidence (demonstration) ofproposition I.The first experiment consists in the setting of a paper withtwo differently painted zones in front of a window with its sides parallelto the horizon. This paper was viewed through a prism which was parallelto the paper. The experiment results in the observation of the followingphenomenon: “the different colors painted on the paper suffer differentdisplacements when viewed through the prism”.

Axiom III (Newton 1730, 18) stipulates that “[a]n Object seen by Re-flexion or Refraction, appears in that place from whence the rays aftertheir last Reflexion or Refraction diverge in falling on the Spectator’s Eye”.From this, Newton concludes that:

Wherefore in both cases the light which comes from the blue half of the paper through theprism to the eye, does in like circumstances suffer a greater refraction than the light whichcomes from the red half, and by consequence is more refrangible. (Newton 1730, 21)

But is the conclusion justifiably inferred from the above premises? Thesecond premise states that the blue zone of the paper is more displacedthan the red one. In order to apply the axiom to this statement we mustfirst ensure that the displacement is exclusively caused by the differentdegrees of refraction of the two color zones. Once this has been done wecould infer that the colors have different refrangibilities. We have to becertain that the cause of the displacement is the refraction, and to that effectwe should eliminate any alternative causes that might have produced thephenomenon. Such a cause might have been one, or any combination, ofthe following: (a) other light, coming from the wall, mingles with the lightcoming from the paper and gives the aforementioned result; (b) the lightcoming from the paper is reflected upon other surfaces, is diffused, andreaches the eye with an angle able to give rise to the discrepancy betweenthe two color zones; (c) the phenomenon may result from the orientationof the prism, which causes the relevant displacement (Lucas raises someof these points in his first letter).

If any one of the above possibilities were the real cause of the phenom-enon, then its explanation would have nothing to do with the differencein refrangibility of rays. Now, we expect Newton to eliminate these al-ternatives to prove his case conclusively. And indeed, describing his firstexperiment Newton writes:

and that the light which fell from the window upon the paper made an angle with the paper,equal to that angle which was made by the same paper by the light reflected from it to theeye. Beyond the prism was the wall of the chamber under the window covered over withblack cloth, and the cloth was involved in the darkness that no light might be reflected fromthence, which in passing by the edges of the paper to the eye, might mingle itself with thelight of the paper and obscure the phenomenon thereof. (Newton 1730, 21)


In the first part shows that Newton ensures that the light coming fromthe paper reaches the eye directly without undergoing any reflections orrefractions. In the second part Newton ensures that light from the wall willnot reach the eye. Thus, Newton makes certain that alternatives (a) and (b)are eliminated. Then, he eliminates possible cause (c).

if the refracting angle of the prism be turned upwards . . . its blue half will be liftedhigher by the refraction than its red half. But if the refracting angle of the prism be turneddownward . . . its blue half will be carried something lower thereby than its red half.

3. CONCLUSION

The discussion of Newton’s first optical paper brings forth the kind ofinferences involved in Newton’s experimental proofs. These are the fol-lowing:

(a) Inductive inferences from the phenomena to the possible properties ofthings that could explain them. These inferences are justified by thefirst two rules of reasoning in the third book of thePrincipia (Newton1729, 398–400) and they involve ‘causal simplification’.

(b) A deduction of consequences from these possible properties, that aretested against the results of controlled experiments.

(c) An elimination of the properties, the consequences of which clashwith the phenomena. The surviving property is deemed to be the truecause.

(d) An inductive generalization (justified by the third rule of reasoning)from the studied case to all similar cases.

Newton’s method of justification can be termed, briefly, as an inductionby elimination. We had the opportunity to stress the differences betweenNewton’s method and the hypothetico-deductive method. This justificatorymethod exemplifies Nickles’ “generative justification” (Nickles, 1988, 40;1989, 299–304), that is, the methodology according to which a theoryis better justified if it can be shown how it was constructed or derivedfrom the background knowledge plus some new experiments. So, Newtondoes not restrict himself to proposing a theory and submitting it to ex-perimental tests. This trait radically distinguishes Newton’s method fromthe hypothetico-deductive method, according to which the warrant for atheory comes solely from the fact that the experimental evidence can bededuced from it. Instead, Newton shows how this theory was derived fromexperiments and some background knowledge (his assumptions).


Furthermore, we see that the “experimental proofs” or “deductionsfromthe phenomena”, as Newton called them, involve much more than simpledeductive (in the standard sense of the term) inferences. They also containampliative inferences, a fact which indicates that Newton used ‘deduction’with a wider meaning than it has nowadays. It must be noted here, that,if Newton considers the list of enumerated alternative explanations to beexhaustive, then the argument by elimination is deductive. We saw, how-ever, that Newton did not think that the experimental proofs are as certainas mathematical demonstrations, and I suggested that this uncertainty mayarise from the possibility that the enumeration of alternatives may not beexhaustive.

Another trait of the method employed by Newton in the first opticalpaper is his refusal to consider causal hypotheses regarding the nature oflight and colors, the “Newtonian positivism”, before he has establishedwith certainty the properties of light. Thus he declines any invitation todiscuss the nature of light and colors, and he refuses to accept any criticismof his theory that is based upon a causal account of light. His methodologi-cal writings make clear that only if experiments could show that his theoryof refrangibility is false, would he be willing to discuss these objections.His correspondence with father Lucas shows, however, that Newton is notalways true to his word. In his first reply, he blatantly refused to discussLucas experiments and dismissed them by inviting Lucas to disregard themand consider, instead, Newton’s crucial experiment. Though in his latercorrespondence with Lucas Newton undertakes the task to criticize Lu-cas’ experiments and defend his theory, he never succeeded in answeringsome of the most important objections of Lucas’. Despite that, Newtonthinks that his theory is free of any recalcitrant evidence and proceeds toconclude that he has proven it experimentally, strongly violating his ownmethodological rules.

I claimed that the Newtoniandeductionincludes induction and otherampliative inferences. It may be objected that Newton contrastsinductionwith deductionand for this reason one should not include the former in thelatter. The motive behind this objection seem to be that Newton uses thesetwo terms differently in the same context, as in the letter to Cotes (Thayer1953, 6) in which Newton states that “principles are deduced from thephenomena and made general by induction”.

The answer to this objection is two-fold. First, Newton did not contrastbetweeninductionanddeduction.In a letter to Oldenburg (Cohen 1958,93), we read: “You know the proper method for inquiring after the proper-ties of things is, to deduce them from experiments”. Upon the completionof the method propositions pertaining to the properties of the things have


been deduced from experiments. Now, in Rule 4 of philosophizing (New-ton 1729, 400) Newton writes that: “In experimental philosophy we areto look upon propositions inferred by general induction from phenomenaas accurately or very nearly true . . . ”. Thus, ‘general induction from thephenomena’ and ‘deduction from the phenomena’ are both used to denotethe method for deriving the properties of things from the phenomena. Thissuggests that Newton did not think that they contrasted with one another.This assumption is reinforced by the following passage from the letter toCotes: “[E]xperimental philosophy proceeds only upon phenomena anddeduces general propositions from them only by induction” (Thayer 1953,7). Here it is clear thatinduction is a kind of thedeductionfrom the phe-nomena. Newton’s usage of these terms is slippery, but this only showsthat Newton shares the same tradition with Descartes, for whominductionwas not a separate form of inference, but a complicated form ofdeduction(AT XI, Rule VII). The modern distinction between these two terms wasnot shared by Newton.

Second, the terminduction was also used with its modern meaning,meaning an ampliative inference from some members of a set to the entireset. This is how Descartes uses it inRule VII of the Regulae, and this ishow Newton uses it in the letter to Cotes. The conclusions that have beendeduced from the phenomena with respect to a particular experimental set-up can be generalized. Thus, they are the universal properties of things.

LIST OF ABBREVIATIONS

ADD: It stands for “Additional Manuscript”, Cambridge University Li-brary.

AT: It stands for the edition of Descartes’ work by Adam and Tannery(Paris: Leopold Cerf, 1897). The Latin numeral indicates the volumeof this edition and the Arabic number(s) the page(s).

CSM: It stands for the translation of part of Descartes’ work in Englishin three volumes by J. Cottingham, R. Stoothoff, and D. Murdoch(Cambridge: Cambridge University Press, 1985).

NOTES

∗ I wish to thank Professors Peter Achinstein and Robert Rynasiewic for reading a draftof this paper, and offering helpful comments, as well as the two anonymous referees whooffered invaluable assistance. I would also like to thank Professor Peter Achinstein for


many stimulating conversations about Newton’s scientific method. Special thanks to Dr.Debbie Brown-Kazazis for making this paper presentable.1 “AT” stands for the edition of Descartes’ work by Adam and Tannery (Paris: LeopoldCerf, 1897). The Latin numeral indicates the volume of this edition and the Arabic num-ber(s) the page(s).2 The term ‘ADD’ stands for ‘Additional Manuscript’ in the Cambridge University Li-brary. The parentheses indicate Newton’s cancellations and interpolations.3 Since some of the terms employed by Newton do not have their modern meaning,I will display them by using italics. These terms are notably ‘deduction’, ‘induction’,‘hypotheses’.4 This alternative possibility will be raised by Lucas in his 1676 letter to Oldenburg (Letter185, Lucas to Oldenburg, 13 October 1676, in W. Turnbull (ed.), vol. 2, 104), with respectto Newton’s experimental crucis.5 This time Newton’s critique is on the mark. As Shapiro remarks (1993, 67), Hooke canavoid the attack only by stipulating that “the initial splitting created cones of light exactlylike those which Newton supposed to have pre-existed in the incident white beam”.6 The “CSM” stands for the translation of part of Descartes’ work in English in three vol-umes by J. Cottingham, R. Stoothoff, and D. Murdoch (Cambridge: Cambridge UniversityPress, 1985).7 Henceforth, when I use the terms “properties” and “causes” in their Newtonian sensediscussed above, I display them in italics. Thus apropertyis an immediate cause, or causeat the first level, whereas acauseis an account of the nature of things, that is, a cause atthe highest level.8 In Rule 4 of thePrincipia explicitlystates that he will not consider as evidence againsthis theory “any contrary hypotheses” (Newton 1729, 400), by which he means, as we haveseen, causal accounts of the nature of light.9 For a more ‘scientific’ account of theexperimentum crucis, and of what Newton actuallyshows with it, see Holtsmark (1970).10 In Rule 4of the Principia, Newton writes that “In experimental philosophy we are tolook upon propositions inferred by general induction from phenomena as accurately orvery nearly true, notwithstanding any contrary hypotheses that may be imagined, till suchtime as other phenomena occur, by which they may either be made more accurate, or liableto exceptions” (Newton 1729, 400).11 The reader will note that in the first paper Newton does not give any ‘technical’ detailsregarding the prisms used, and, in general, regarding the experimental conditions. Thislapse gave rise to various difficulties, as others who tried to repeat his experiments observedmostly different things. As a result Newton, in his later correspondence, started reportingthe conditions of his experiments with increasing details. For a thorough discussion of thisissue see Schaffer (1989).

REFERENCES

Achinstein, P.: 1990, ‘Newton’s Corpuscular Query and Experimental Philosophy’, in P.Bricker and R. I. G. Hughes (eds.),Philosophical Perspective on Newtonian Science,The MIT Press, Cambridge MA, pp. 135–175.

Bacon, F. : 1990,Novum Organum, translated and edited by P. Urbach and I. Gibson, OpenCourt, Chicago.


Brody, B. A. and R. E. Grandy (eds.): 1989,Readings in the Philosophy of Science,Prentice-Hall, NJ.

Cartwright, N.: 1983,How the Laws of Physics Lie, Oxford University Press, Oxford.Cohen, I. B. (ed.): 1958,Isaak Newton’s Papers and Letters on Natural Philosophy,

Harvard University Press, Mass.Cottingham, I. and R. Stoothoff, and D. Murdoch, D. (eds.): 1985,The Philosophical

Writings of Descartes, Cambridge University Press, Cambridge.Curd, M. V.: 1980, ‘The Logic of Discovery: An Analysis of Three Approaches’, in T.

Nickles, (ed),Scientific Discovery, Logic and Rationality, reprinted in B. A. Brody andR. E. Grandy (eds.) 1989:Readings in the Philosophy of Science, Prentice-Hall, NJ, pp.417–430.

Descartes, R.: 1896,Oeuvres, Edited by C. Adam and P. Tannery in 12 vol. Paris.Descartes, R.: 1966,Discours de la Methode suiw d’ Extraits de la Dioptrique, des

Meteores et de Lettres, Garnier-Flammarion, Paris.Gooding, D.: 1990,Experiment and the Making of Meaning, Kluwer Academic, Dordrecht.Hall, A. R. and Marie Boas Hall (eds.): 1962,Unpublished Scientific Papers of Issac

Newton, Cambridge University Press, Cambridge.Hall, A. R.: 1993,All Was Light: An Introduction to Newton’s Opticks, Clarendon Press,

Oxford.Holtsmark, T.: 1970, ‘Newton’s Experimentum Crucis Reconsidered’,American Journal

of Physics38(10), 1299–1235.McGuire, I. E.: 1966, ‘Body and Void in Newton’sDe Mundi Systemate: Some New

Sources’,Archive for History of Exact Science3, 238–9.McGuire, I. E.: 1970, ‘Newton’s Principles of Philosophy: An Intended Preface for the

1704 Opticks’,The British Journal for the History of Science5(18), 178–186.McGuire, I. E. and M. Tamny: 1983,Certain Philosophical Questions, Cambridge

University Press, Cambridge.McMullin, E.: 1978, Newton on Matter and Activity, University of Notre Dame Pres,

Indiana.Newton, I.: 1729,Principia, University of California Press, Berkeley, 1962.Newton, I.: 1730,Opticks, based on the fourth edition, Dover, New York, 1952.Nicles, T.: 1988, ‘Reconstructing Science: Discovery and Experiment’, in D. Batens and I.

P. Van Bendegem (eds.), Theory and Experiment: Recent Insights and New Perspectiveson Their Relation, Dordrecht: Reidel, pp. 33–53.

Nickles, T.: 1989, ‘Justification and Experiment’, in D. Gooding, T. Pinch and S. Schaffer(eds.),The Uses of Experiment, Cambridge University Press, Cambridge, pp. 299–333.

Schaffer, S.: 1989, ‘Glass Works: Newton’s prisms and the uses of Experiment’, in D.Gooding, T. Pinch and S. Schaffer (eds.),The Uses of Experiment: Studies in the NaturalSciences, Cambridge University Press, Cambridge, pp. 67–104.

Schuster, I. A.: 1986, ‘Cartesian Method as Mythic Speech: A Dichronic and StructuralAnalysis’, in I. A. Schuster and R. R. Yeo (eds.),The Politics and Rhetoric of ScientificMethod, Kluwer, Reidel, pp. 33–95.

Shapiro, A. (ed.): 1983,The Optical Papers of Newton, Cambridge University Press,Cambridge.

Shapiro, A.: 1993,Fits, Passions, and Paroxysms, Cambridge University Press, Cambridge.Thayer, H. S. (ed.): 1953,Newton’s Philosophy of Nature, Hafner Press, New York.Turnbull, H. W. (ed.): 1959,The Correspondence of Isaak Newton, 3 vol. University Press,

Cambridge.

https://www.researchgate.net/publication/243755077_Experiment_and_the_Making_of_Meaning?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/243487656_Newton's_Experimentum_Crucis_Reconsidered?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/243487656_Newton's_Experimentum_Crucis_Reconsidered?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/275846486_The_Correspondence_of_Isaac_Newton?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/275846486_The_Correspondence_of_Isaac_Newton?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/226431813_Body_and_void_and_Newton's_De_Mundi_systemate_Some_new_sources?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/226431813_Body_and_void_and_Newton's_De_Mundi_systemate_Some_new_sources?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/231961858_Newton's_Principles_of_Philosophy_An_Intended_Preface_for_the_1704_Opticks_and_a_Related_Draft_Fragment?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/231961858_Newton's_Principles_of_Philosophy_An_Intended_Preface_for_the_1704_Opticks_and_a_Related_Draft_Fragment?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/241455909_All_Was_Light_An_Introduction_to_Newton's_Opticks?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/241455909_All_Was_Light_An_Introduction_to_Newton's_Opticks?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/275842539_Isaac_Newton's_Papers_and_Letters_on_Natural_Philosophy?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/275842539_Isaac_Newton's_Papers_and_Letters_on_Natural_Philosophy?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/273108079_The_Philosophical_Writings_of_Descartes?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/273108079_The_Philosophical_Writings_of_Descartes?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==

https://www.researchgate.net/publication/283420415_How_the_Laws_of_Physics_Lie?el=1_x_8&enrichId=rgreq-3dbf17c6163a2ac521af4ca92f88df9b-XXX&enrichSource=Y292ZXJQYWdlOzIyNjE5NjQyMztBUzoxMDQ4NDc4MzE0NzAxMDBAMTQwMjAwOTA3OTkwNg==


Westfall, R. S.: 1966, ‘Newton Defends his First Publication: The Newton–Lucas Corre-spondence’,Isis57(3), 299–314.

Manuscript submitted November 25, 1996Final version received December 12, 1998

Department of Educational SciencesUniversity of CyprusKallipoleos G1P.O. Box 20537Nicosia 1678Cyprus

Newton's Experimental Proofs as Eliminative Reasoning

Documents