On the Conceptual and Civilization Frames in René Descartes’ Physical Works

doi:10.4236/ahs.2013.23015

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Advances in Historical Studies

2013. Vol.2, No.3, 106-125

Published Online September 2013 in SciRes (http://www.scirp.org/journal/ahs) http://dx.doi.org/10.4236/ahs.2013.23015

106

On the Conceptual and Civilization Frames in René Descartes’

Physical Works

Paolo Bussotti1, Raffaele Pisano2

1Research Centre for the Theory and History of Science, University of West Bohemia, Pilsen, Czech Republic

2Sciences, Sociétés, C ultures Dans Leurs Évolutions, University of Lille 1, Lille, France

Email: paolobussotti66@gmail.com, pisanoraffaele@iol.it

Received April 15th, 2013; revised May 20th, 2013; accepted May 30th, 2013

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

The paper try to provide a contribution to the scientific—historiographic debate concerning the relations

between experiments, metaphysics and mathematics in Descartes’ physics. T he three w orks o n which the

analysis is focused are the Principia philosophiae and the two physical essays: La Dioptrique and Les

Météores. The authors will highlight the profound methodological and epistemological differences char-

acterizing, from one side, the Principia and, from the other side, the physical essays. Three significant

examples will be dealt with: 1) the collision rules in the Principia philosophiae; 2) the refraction law in

La Dioptrique; 3) the rainbow in Les Météores. In the final remarks these differences will be interprete d

as depending upon the different role Descartes ascribed to the three books inside his whole work. The

concepts of intensity and gradation of the physical quantities used by Descartes will provide an important

interpretative means. In this paper, we compare the aprioristic approach to physics typical for Descartes’

Principia with the experimental and mathematical one characterizing Descartes’ Essays.

Keywords: Descartes; Newton; Collision Rules; Refraction Law; Rainbow; Intensity and Gradation of the

Physical Quantities; Science and Society in the XVII Century

An Outline

On Science & Society. The social and civilization environ-

ment in which a scientist lives has profound influences on the

way how his scientific results and methods are framed (e.g. see

Schuhl). This is specifically true for the 17th century, the epoch

of the scientific revolution and a century of deep social and

political transformations. Nevertheless, we think influence of

the social-political situation on the work of a scientist has to be

deduced directly from the analysis of his scientific works. In

other terms: an analysis of the society in a certain period can be

useful to understand the general direction taken by the science

in that period, but, in itself, it is not enough to understand the

specific work and results of a certain scientist. This kind of

general analysis risks to become a sort of an a priori passe-

partout through which the scientific work is analysed and risks1

to induce serious misunderstandings on the way in which a

certain scientist presented the results of his researches. It is

always necessary to begin a historical research—also a research

concerning the relations between science and society in a deter-

mined period—from the alive, both theoretical and technical

work of the scientists. If, in the analysis of the whole work of a

scientist, the historian of science reveals some unclearness or

internal inconsistencies or a lack of coherence between the

methods used by this scientist in different works of his and if

all these questions cannot be explained either with technical

problems (for example the lack or the misunderstanding of

certain mathematical methods) or with the general methodo-

logical and epistemological convictions of the scientist himself,

then it is necessary to think of the general structure of the soci-

ety in that period. Therefore technical analysis of the results

and methods used by the scientist is a priori considered and

then evaluated within civili za tion.

On Science. The case of René Descartes (1596-1650) is em-

blematic in this sense: in his essays La Dioptrique and Les

Méthéores Descartes proposes—among other results—his the-

ory of refraction and of rainbow. Every passage of these two

works can be explained taking into account: 1) the level of the

science in the 17th century; 2) Descartes’ experiments and

methods; 3) Descartes’ use of mathematics; 4) Descartes phi-

losophical convictions (Hattab, 2009). These books could be

understood without taking into account the social non-scientific

context in which Descartes lived. The situation as to the Prin-

cipia philosophiae is different at all: we will see in the final

remarks of this paper that many results and argumentative

structures exposed by Descartes in his Principia can be explained

taking into account Descartes’ epistemological and philosophical

convictions, but other parts of the book and some reasoning that

appear tormented and unclear can be clarified only considering

the particular social situation in which Descartes lived and op-

erated. The sociological analysis becomes hence interesting and

can represent a great means to understand the evolution of the

scientific ideas only if it is based on the examination of the

theoretical-technical results obtained by the scientists and ex-

plained in their works. The case Newton and his civilization

science (Buchwald & Feingold, 2011) is as interesting as the

1On that see a good essay by Buchwald and Feingold (Buchwald & Fein-

gold, 2011 ).

P. BUSSOTTI, R. PISANO

one of Descartes: the different social context in which Newton

lived allowed him a major freedom than Descartes’ (see final

remarks of our paper). But in this case, too, the examination

becomes interesting basing on Newton’s physical and mathe-

matical works. Only in this manner the sociological analysis of

science becomes perspicuous and useful for history of science

and scientific concepts.

Isaac Newton (1642-1727) explicitly claimed that a model of

the solar system had to show the positions held by a planet

(Jupiter in the specific case mentioned in the forthcoming quo-

tation) in the course of time, and that this condition cannot be

fulfilled following Descartesian physics. Consequently the

physical system described by René Descartes (1596-1650) in

his Principia Philosophiae (1644; see Figures 1 and 2) is not a

good model of the universe. In his unpublished work De

Gravitatione et aequipondio fluidorum (Ruffner), Newton criti-

cized the model of the solar system proposed by Descartes in

his Pincipia philosophiae as follows:

And hence, about the place of Jupiter, which it kept the

year before, and with equal reason, about the prior place

of a moving body anywhere, according to the doctrine of

Descartes [illeg] it is manifest that not even God himself

(standing newly established with things) could accurately

and in a geometrical sense describe [it], especially when,

on account of the changed positions of bodies, it would no

longer exist in the nature of things2.

Figure 1.

The frontispiece of the first edition of Descartes’ Prin-

cipia (1644)3.

Figure 2.

The frontispiece of the second edition with General Scholium by New-

ton’s Principia (1713)4.

Albeit, from an epistemological point of view, it is difficult

to exactly identify all characteristics a descriptive-explicative

model of physical phenomena should keep, some of them can-

not be ignored. Two of these cha rac ter ist ics ar e:

1) The coherence of the principles that are at the basis of the

system, that is the principles must not be mutually contradic-

tory.

2) The possibility to determine quantitative relations between

the sizes of the system.

We note that in the classical physical studies, the possibility

to express the position of a body as a function of the time, is

necessarily a law of motion fundamental for quantitative rela-

tions. Generally speaking in order to express such law, it is

necessary to determine a physical system in which the space-

variable can be decomposed into three (dimensional) mutually

perpendicular directions5. Then, for every motion, the position

of the moving body can be expressed in function of the time i.e. x

= f(t), y = g(t), z = h(t). Thus, a law of motion can well interpret

a classical physical phenomenon if a Descartesian system (time

and each of the three directions) is provided. Hence, time and

space have to be uniform quantities as far as they are the bases

of the reference systems. A position of a body is a function of

2“Et proinde de loco Iovis quem ante annum habuit, parique ratione de

præterito loco cujuslibet mobilis manifestum est juxta Cartesij [illeg] doc-

trinam, quòd ne quidem Deus ipse (stante rerum novato statu) possit accu-

ratè et in sensu Geometrico describere, quippe cùm propter mutatas cor-

porum positiones, non ampliùs in rerum naturâ existit” (Newton folios 9,

Ms Add. 4003, Cambridge University Library, Cambridge, UK [retrieved via:

http://www.newtonproject.sussex.ac.uk/view/texts/normalized/THEM00093]).

3Descarte s 1897-1913, X-2.

4The English tra nslation (1729) was by Andrew Motte (1696-1734) found

in the second Latin edition (171 3).

5That is in modern terms as rectangular coordinate systemalso called

escartesian or Cartesian coordinate system by three functions for coordi-

nates.

P. BUSSOTTI, R. PISANO

108

time, but the space itself is not. According to Newton6 the fun-

damental problem of Descartes’ physics can be so summarized

(see Figure 3):

The essays La Dioptrique, Les Météores, La Géométrie (see

Figures 4 and 5) and numerous letters (Descartes, 1897-1913,

I-II-III-IV-V) provide the idea of a completely different Des-

The hypotheses of Vortices is

pressed with many difficulties.

That every Planet by a radius

drawn to the Sun may describe

areas proportional to the times of

description, the periodic times of

the several parts of the Vortices

should observe the duplicate

proportion of their distances from

the Sun. But that the periodic

times of the Planets may obtain

the sesquiplicate proportion of

their distances from the Sun, the

periodic times of the parts of the

Vortex ought to be in sesquipli-

cate proportion of their distances.

That the smaller Vortices may

maintain thei r lesser revolutions

about Saturn, Jupiter, and other

Planets, and sw im quietly and

undisturbed in the greater Vortex

of the Sun, the periodic times of

the parts of the Sun’s Vortex

should be equ al. But the rotation

of the Sun and Planets about their

axes, which ought to correspond

with the motions of their Vo r-

tices, rece d e far from all these

proportions. The motions of the

Comets are exceedingly regular,

are govern’d by the same laws

with the motions of the Pl anets,

and can by no means be ac-

counted for by the hypotheses of

Vortices. For Comets are carry’d

with very eccentric m ot ions

through all parts of the heavens

indifferently, with a fre edom that

is incompatible with the notion of

a Vortex. [...]

a) If—as it is the case in Des-

cartes—the space is identified

with the res extensa, that is, if

the separation between space

and bodies m oving in the space,

is substantially denied, then

the space has the same char-

acteristics of the moving bod-

ies and the position of the

space itself becomes a func-

tion of time. Therefore it can

happen that a point existing at

the instant 0

t, does not exist

anymore at the instant

tt , so that a system of

coordinates in which the posi-

tions of the bodies can be

given, cannot be establis hed.

b) Newton writes that in Des-

cartes’ system not even a God

could determine the position

of a planet as a function of

time and in De gravitatione he

explains in detail the reasoning

we have summarized in a

modern language.

c) Thus, according to Newton, the

description of th e physi cal wor ld

ideated by Descartes in his Prin-

cipia Philosophiae (hereafter

Principia) does not satisfy the

two characteristics needed for

a model.

d) Besides thes e, ther e are fur ther

problems as the consequences

of some laws expressed in the

Principia and contradicted by

the exper ien ce (as it is th e case

of the collision rules between

two bodies) or the unscrupu-

lous resort to analogy and the

lack of cl earnes s as to the r ela-

tions between experience-ex-

periment and theory.

Figure 3.

Newton’s first paragraph on (implicitly) Descartes at the beginning of

the General Scholium7.

cartes. He supplied substantially correct modelling of phenom-

ena, as the refraction (Ivi, La Dioptrique, discours II, VI) with

the consequent genial explanation of the rainbow and of other

optical effects (Ivi, Les Météores, discours VIII, VI). Some-

times analogy brought him to incorrect explanations, as it is the

case for the origin of the colours (Les Météores, discours VIII,

VI). However, in these cases, too, a profound attempt to make

the theory coherent with the facts is present. The idea to meas-

ure and to quantify the sizes constitute the conceptual and

methodological basis of La Dioptrique and of Les Météores

even if the transcription into mathematical terms is not always

explicit. Particularly La Géométrie (Ivi, VI) deserves a separate

series of considerations: despite mathematical problems are

dealt with (hence not directly connected with the knowledge of

the external world), the new modelling proposed by Des-

cartes—the analytical geometry—will be fundamental for sci-

ence, too, because of the idea to transcribe geometrical data into

an analytical form. The Essays and some letters arouse hence a

different impression from that given by the Principia.

In the La Dioptrique (and Les Météores) he was able to pro-

vide—plausible, even if non always exact—early models of the

phenomena as refraction, rainbow and origin of the colours

considering empirical data and framing them into a theoretical

structure, as it will be clarified in the third section of our paper.

Differently from this approach, in the Principia, as well known,

Figure 4.

The frontispiece of Discours de la m éthode (1637)8.

6It is well k nown that Newt on spok e of absolu te time and absolut e space in

the general General Scholium (Newton, [1713] 1729) where it does not

begin with the introduction of the concepts of absolute space and absolute

time, but with the prove that the vortices-theory of Descartes is untenable.

Likely Newton introduced explicitly his concepts of absolute space and

time as an epistemological answer to Descartes’ theory. In this manner the

initial part of the General Scholium can be interp reted as the ph ysical refu -

tation of Descartes’ theory and the second part as the epistemological refu-

tation . On historical- philosophical co nceptualizati on around Newtoni an co-

lour theory and the new analytical theories one can see Panza (Panza, 2005,

2007), Blay (Blay, 1983, 1992, 2002), Rashed (Rashed) and on Newtonian

Optik Hall (Hall, 1993; see also Halley,1693). On Fresnel’s optic one can

also see Rosmorduc, Rosmorduc and Dutour (Rosmorduc J, Rosmorduc V,

Dutour) interes ti ng for our ai m s.

7Newton, [1713] 1729: p. 387. Recently on Newton a critic French edition

is remark able (Panza, 2004).

8Descartes 1897-1913, VI. Discours de la Méthode (Ivi: pp. 1-79). It in-

cludes La Dioptrique (Ivi: pp. 80-228), (Ivi, Les Météores: pp. 231-366),

Géométrie (Ivi: pp. 367-485). Le Monde (Ivi, XI-1: pp. 3-118). For the Latin

edition (1644) of the Principia see Ivi, VIII-1; for the French translation

(1647) see: Ivi, IX-2.

P. BUSSOTTI, R. PISANO

Figure 5.

The first page of the La dioptrique (1694)9.

Descartes tried to supply a global physical theory looking for

its foundation in few basic notions without resorting to any

quantification. He limited his speeches with qualitative and

analogical arguments. Descartes does not seem to fully catch

the difficulty and complexity of some problems as the nature of

gravity and of magnetism (Le Monde ou Traité de la lumière

(hereafter Le Monde), Descartes, 1897-1913, XI; Id., Principia,

IX-2, Part IV, § 20-27: pp. 133-183). The example of gravity is

particularly significant: Descartes’ mechanisitic conception

brought him to think that the origin of gravity (to consider as a

phenomenon taking place on the earth) depends on the effects

of the quick movement of the particles (“particulae”) of the

second element around the earth (Principia, 1644, VIII, Part IV,

§ 20-21)10 (see Figures 6 and 7).

The earth itself and the bodies on the earth are mostly com-

posed of particles belonging to the third element. They are

heavier than those of the second element surrounding the earth.

The movement of the particles of the second element exerts a

pressure on the bodies composed by particles of the third ele-

ment so that they tend to the centre of the earth. In synthesis

this is the mechanistic conception of gravity exposed by Des-

cartes. A consequence of this conception is the theoretical im-

possibility to determine a relation between mass as physical

measurable quantity and quantity of matter as (classical Des-

cartesian) conception of internal part of an object (see Figure 8).

A consequence is that the explanation between what is the

mass (physical measure) and what is the quantity of matter

(mathematical interpretation) was not easily identifiable due

Figure 6.

Descartesian gravity and magnetism.

Figure 7.

Descartesian gravity and magnetism11.

9Descarte s, 1897-1913, VI.

10As to th e theory o f the par ticles compos ing th e three ele ments (Descart es,

1897-1913 [Principia, 1644, III, § 48-53] VIII-1: pp. 102-107) and in par-

ticular the chapter 52 (Descartes 1897-1913, VIII-1: p. 105, line 11-30)

titled Tria esse huius mund i aspedabilis elementa.

11Figure 6: Descartes 1897-1913 [Principia, 1644, VIII-1, Part IV, § 20-21]

IX-2: pp. 21 0-211 [Ful l Latin version: Ivi, VIII-1: pp. 1-348].

P. BUSSOTTI, R. PISANO

110

It can happen i.e. that, albeit a mass of gold is twenty times

heavier than a quantity of water of the same size, it does not

contain twenty times the quantity of matter contained in that

mass of water, but only four or five times […] 12

Figure 8.

Some Descartes’ argume n t s on matter concept13.

their difficulties of transcription into quantitative physical terms.

The mechanistic and a priori conviction of Descartes brought

hence him to the impossibility to have a well defined concep-

tion of space and of mass14. This is a substantial, not only for-

mal difference. In fact, the scientific framework of the treatises

can deceive. For example, in Principia, Newton wrote eight

definitions and the three laws (or axioms) at the beginning (see

Figure 9).

Therefore one can get the impression he started from these to

explain the phenomena analysed in the three books of Principia.

Actually, the two introductory sections (definitions and axioms)

give an Euclidean order to the text that is different from the

way in which Newton reached to determine the nature of the

phenomena. The definitions and the laws were enucleated on

the basis of the phenomena, not before a detailed examination

Figure 9.

Newton’s laws15.

12“Et fieri potest, ut quamvis, exempli caussa, massa auri vicies plus pon-

deret, quam moles aquae ipsi aequalis, non tamen quadruple vel quintuplo

plus materiae terrestris contineat […]” (Descartes [Principia, 1644, Part IV:

p. 202] VII I-1: p. 213, line 16). The translation is ours.

13Ibidem.

14The concept of mass from physical and mathematical standpoint was a

hard concept until 19th century for new theories i.e. like chemistry and

thermodynamics, machines theory (Pisano, 2010, 2011). For example

Lazare Carnot (1753-1823) explicitly was ambiguous (Gillispie &Pisano

2013: p. 377) on the concept of force (Carnot, 1803: p. xj, p. 47) and mass

assuming both of Descartesian and Newtonian assumptions (Carnot, 1803:

p. 6). Ernst Mach (1838-1916) wrote interesting speeches on that (Mach,

([1896] 1986): pp. 368-369) tried to formulate an operative interpretation o

mass using the third principle of mechanics (Mach, 1888, [1896] 1986)).

15“Axioms; or Laws of Motion. Law I. Every body perseveres in its state o

rest, or of uniform motion in a right line, unless it is compelled to change

that state by forces impressed thereon; Law II: The alteration of motion is

ever proportional to the motive force impressed; and is made in the direc-

tion of the right line in which that force is impressed; Law III: To every

action there is always opposed an equal reaction: or the mutual actions o

two bodies upon each other are always equal, and directed to contrary

arts.” (Newton, [1686-7] 1803, I: pp. 19-20; Italic style and capital letters

belong to the author). (Newton, [1686-7] 1803, I: p 2; author’s italic style

and Capital letters). On forces and their geometrical interpretation one can

see De Gan dt (De Gandt, 1995).

P. BUSSOTTI, R. PISANO

and comprehension of the phenomena themselves. Instead in

Descartes’ Principia the laws, and above all the ideas concern-

ing the constitution of matter, were thought almost independ-

ently from phenomena and, afterwards, applied to them.

On the Principia Philosophiae

In this section, we will deal with the cases in which the

physical laws established by Descartes in his Principia are

self-contradictory and contradicted by the experience itself,

particularly on the collision rules theory.

Some Historiography on Descartes’ Collision Rules

The historiography concerning Descartes’ collision rules is

conspicuous. Here we analyze only those studies directly con-

nected with the logic of our reasoning16.

Ernst Cassirer (1874-1945) stresses that the collision rules

are self-contradictory, even if he does not enter into details.

Consequently such rules do not provide a unified picture and,

hence, a model of the phenomenon. Cassirer ascribes this situa-

tion to the fact that Descartes

[…] leaves the continuous and patient development of his

deductive-mathematical premises and passes directly to

explain concrete particular phenomena that are very com-

plex17.

Nevertheless, it is necessary to point out that in other cases of

complex phenomena, as the refraction and the rainbow, Des-

cartes is faithful to the mathematical approach of his own work.

Pierre Boutroux (1880-1922) after having eulogized Des-

cartes for the introduction of the inertia principle and the con-

servation of the quantity of motion principle, writes as to the

collision: “Unfortunately, Descartes makes a very serious mis-

take, that is surprising from his part”18. The mistake consists in

the fact that Descartes did not catch the vectorial nature of the

quantity of motion. The mistakes in the collision rules are due,

according to Boutroux, to this misconception.

René Dugas (1897-1957) claims there is more than one rea-

son why Descartes did not succeed in the explanation of the

collision: a) lack of distinction between elastic and inelastic

collisions (Dugas, [1954] 1987: pp 150-151); see also 1954; b)

existence of dissymetries with regard to the reasons that can

produce, increase or diminish the quantity of motion of a body;

c) lack of comprehension of the vectorial nature of velocity

(Ivi). Dugas adds that the experience is anyway necessary for a

correct formulation of the collision rule (Ivi). Furthermore he

underlines that from Descartes’ correspondence, it is possible to

deduce he had carried out some experiments, but that, between

experimental results and principles, he had chosen the princi-

ples. Therefore Dugas ascribes the failure of Descartes’ colli-

sion rules to an unclear comprehension of the basic principles

connoting the motion quantity (theoretical reason) and to the

lack of serious experiments on this subject (empirical reason).

Pierre Costabel (1912-1989), after having analysed the colli-

sion rules in Descartes claims:

It has been said and repeated the Descartesian collision

rules are only an outline. Nevertheless, it has been stress-

ed that the principles, of which such rules would be an

outline, were already acquired in Descartes’ thought. Ac-

tually we believe that things work in the opposite manner.

These rules are only an outline because they are the ex-

pression of a thought that is still researching19.

By the way, still Costabel’s opinion that Descartes proposed

only some outlined rules in the work he considered the result of

his most mature thought in physics appears disputable.

Recently, Stephen Gaukroger discusses that Descartes’ phys-

ics is based upon modelling drawn from statics and tries to

explain the genesis itself of the collision rules on this basis. He

proposes an interesting examination of the fourth rule (Gauk-

roger, 2000: pp. 60-80).

Peter McLaughlin, analyses the Descartesian concept of de-

termination of a motion. He also frames the Descartesian rules

inside a context deriving from statics and, in this way, he tries

to provide an explanation of such rules (McLaughlin, 2000: pp.

81-112).

Beyond the principles explicitly formulated in his works,

likely Descartes also resorted to some principles of minimum

exposed in some of his letters. Gary C. Hatfield mentions a

letter on 17 February 1645 to Clerselier20 (1614-1684) in which

Descartes wrote:

[…] when two bodies in incompatible modes collide,

some change in these modes must truly occur, so as to

render them compatible, but that this change is always the

least possible […]21.

McLaughlin also points out that Descartes resorts to a “prin-

ciple of minimal modal change” (McLaughlin, 2000: p. 99). He

also tries to interpret the meaning of mode. In particular, he

remarks that determination and velocity of a motion are two

different modes. It is then maybe possible to think that the col-

lision rules are conceived so that the modal change is the less

possible. In this manner, for example, in the rule 4, the change

of the determination of the body B represents a modal change

less than the one existing if the body C, too, would move, be-

cause, in this case, two modes would change: determination and

velocity.

The existence of principles of minimum in Descartes’ corpus

is reasonable by what he wrote in the fifth discours of the Les

Météores concerning with form of the clouds under the action

of irregular winds:

[...] figure which can least [assume the form and] prevent

16With regard to the various factors on which historiography of science

depends, see: Kragh, 1987; Pisano & Gaudiello, 2009a, 2009b; Kokowski,

2012; Poincaré, [1923] 1970, [1935] 1968; Rossi; Taton, 1965, 1966;

Westfall, 197 1.

17“[…] er den stetigen Gang und den geduldigen Ausbau seiner deduk-

tivmathematischen Voraussetzungen verläßt, um unvermittelt zu der Erk-

lärung verwickelter konkreter Sonderphänomene”. (Cassirer, [1906] 1922:

p. 479). The translation is ours.

18“Malheureu s ement, Descartes commet u n e er r eur très gr ave et qui est b i en

surprenante da sa part”. (Boutroux, 1921: p. 67 7) . The translation is ours.

19“On a dit et redit que les règles cartésiennes du choc ne sont qu’une es-

quisse, mais on l’a fait en sous-entendant que les principes dont elles se-

raient l’esquisse étaient déjà fermes dans la pensée de Descartes. La réalité

nous parait différente. Ces règles ne sont qu’une esquisse parce qu’elles

sont l’expression d’une pensée en état de recherche”. (Costabel,[1967]

1982: pp. 141-[152]158). The translation i s ours. See also Costabel, 1960.

20Clerselier is an important figure in the scientific frameworks of Descartes.

He edited and translated many Descartes’ works i.e. Correspondences

(1657, 165 9, 1667), Le Mon de (1667) and Principes (1681).

21“[…] lors que deux cors se rencontrent, qui ont en eux des modes income-

atibles, il se doit véritablement faire quelque changement en ces modes,

our les rendre compatibles, mais (que) ce changement est tousiours le

moindre qui puisse être […].” (Descartes, 1897-1913, IV: p. 185, line 13).

Author’s italic. See also Hatfield, 197 9, p. 133.

P. BUSSOTTI, R. PISANO

112

[opposes less resistance to] their movement [...]22

The plurality of approaches through which many distin-

guished scholars and historians tried to explain the reasons that

led Descartes to formulate collision laws that are self-contra-

dictory and not confirmed by experience, proves that this ques-

tion is not clear at all. Therefore, every explanation has, at least

partially, a conjectural and hypothetical character.

Collision Dynamics and Physics-Mathematics

Arguments

The study of the collision rules between two bodies is a sub-

ject on which the literature was relatively abundant in Des-

cartes’ age (McLaughlin, 2000: pp. 81-112). Descartes estab-

lished seven rules (Descartes, 1897-1913, Principia, VIII-1: pp.

69-69; see Figure 10).

Descartes did not make distinction between elastic and in-

elastic collision. However, considering the structure of his rea-

soning above exposed, he was referring to elastic collisions on

a surface without friction. It is known that only the first one of

these previous rules is correct. A part from this, do these rules

have an inner coherence—for inner coherence we mean the

property according to which no contradictory conclusions can

be draw from the principles? Let us see an example in order to

consider two bodies B and C, the first one of mass 5, the second

one 7.

Let C be at rest and let B move with velocity v. After the col-

lision, C remains at rest, while B inverts the direction of its

motion (by using rule 4).

Let us now imagine that B increases its mass with continuity

and that C decreases it with continuity, in a way that the sum of

the sizes remain invariable. When B has size 7 and C size 5,

1) If two bodies B and C, whose mass23 is equal, go one against the other with the same speed, then, after the collision, they bounce back in the starting

direction with unmodified speed; 2) In the same situation, but with B greater than C, the body C, after the collision, bounces back in the original direc-

tion and the two bodies proceed unif ied in th at dir ectio n; 3) I f B and C have t he same size, b ut B i s qu icker than C, t hen, after the collision, C bounces

back and, mutatis mutandis, the situation is t he s ame as i n t h e ru le 2 ; 4 ) If C is bi gger th an B and C is at res t, wh atev er th e sp eed of B is af ter th e colli-

sion, C remains at rest and B bounces back in the direction from which it was coming; 5) If C is smaller than B and C is at rest, when B collides with C,

the two bodies proceed unified, according to the principle of conservation of the quantity of movement; 6) If C is at rest and B and C have the same

size, and if B hits C, after the collision, C will move in the same direction and verse as B, while B itself bounces back; 7) If B and C move in the same

verse and C is bigger and slower than B and the excess of the velocity of B is greater than the excess of the size of C, then B transfers part of its

movement to C, so that the two bodies move with the same velocity in the same verse. The rule also considers the symmetric case in which the excess

of speed of B is less than the excess of size of C.

Figure 10.

Collision rules arguments24.

22“[...] la figure qui peut le moins empêcher leur mouvement […]”. (Descartes, 1897-19 13 , VI: p. 286, lines 28-29). Th e translation is ours.

23The word u sed by Desc artes f or mass is—in general—“mole” (Descartes, 1897-1913, VIII: p. 68, line 9) when he speaks of the third rule of the collision and

other o ccu rrences. In t he Principia Descartes uses the word “corpus” plus an adjective (“major” or “minor”). For example, at the beginning of the fifth collision

rule, we read: “Quinto, si corpus quiescens C esset minus quam B […]” (Descartes, 1897-1913 Principia VIII:p. 69, line 1). We have translated th ese words

with mass because o th er tr ansl ati ons wou ld b e even wor se an d wi th out ref ereei ng to moder n co n cep t o f t he mas s. On t he h is tor y of th e co ncept of mass, at first

glance one cans see Jammer (Jammer, 1961).

24Descartes, 1897-1913, VIII: pp. 68-69.

P. BUSSOTTI, R. PISANO

after the collision, B pushes C and the two bodies prosecute

their motion unified (rule 5).

Thus, because of the continuity principle, a physical state

necessarily exists in which, after the collision, B remains at rest

and this must happen when B is as great as C.

The conclusion of this reasoning, deduced by applying the

rules 4 and 5, is that when a body B of size m and speed v

strokes a body C—that is at rest—of the same size, B remains

at rest and C prosecutes in the same direction and verse as B

and with speed v.

However, this results contradicts the rule 6.

Beyond the lack of inner coherence, there is also the problem

that the Descartesian rules of the collision contradict the daily

experience concerning the collisions themselves in an evident

manner. In the Principia, Descartes—who was aware of this—

underlined (Descartes, 1897-1913, IX-2, §53) that his rules are

referred to ideal situations that can be hardly experimented,

after having concluded the paragraph 52 claiming that “these

rules [of collision] are so evident that no empirical confirmation

is necessary” 25. In this part of his scientific framework the rela-

tion between experience and modelling-theory would provide

that i.e. single events of a phenomenon are determined by cir-

cumstances that are contingent in respect to an a priori theo-

retical model. Thus this kind of approach is typically deductive

and aiming to test a final theoretical reasoning. As a matter of

fact, the eventual inconsistency between theory and experience

depends either on such circumstances or on the inadequacy of

the model to represent the phenomenon to which it was ap-

plied26.

The lack of an agreement between experiences-data and

modelling-theory is typical of a scientific theory, especially

physics and chemistry. For example, a unit of measurement is

effectively a standardised quantity of a physical (and chemical)

property, used as a factor to express occurring quantities of that

property. Therefore, any value of a physical quantity is ex-

pressed as a comparison to a unit of that quantity. In the physics

mathematics27 domain one generally precedes by means of

calculations, therefore the units of measurement are not a prior-

ity in terms of a solution to an analytical problem (Pisano, 2013;

Lindsay, Margenau, & Margenau). In this sense, the physical

(and chemical) nature of the quantities is not a priority28. One

may discuss the role played by a certain science in history (e.g.,

physics), focusing solely on the historical period, the kind of

mathematics adopted and the relations between experiments

and theory in the analysed historical period (Pisano, 2011). For

our aim, the most important aspect is the role played by the

relationship between physics and mathematics adopted in a

scientific theory in order to describe mathematical laws—e.g.,

the second Newtonian mathematical law of motion or, in the

case of Descartes’s Principia, the lack of such mathematical

structure and its conesquences (Nagel, 1961, 1997). On the

other hands, the time is a crucial physical magnitude in me-

chanics (Truesdell, 1968) but in the aforementioned, the time

(and space) is also a mathematical magnitude since it is a

mathematical variable in variations (later derivatives) opera-

tions aimed to interpret a certain phenomenon. Most impor-

tantly, if we lose the mathematical significances of time and

space magnitudes, we would lose the entire mechanical para-

digm. Nevertheless, the approaches to conceive and define

foundational mechanical-physical quantities and their mathe-

matical quantities and interpretations change both within a

physics mathematics domain and a physical one (Duhem). One

could think of mathematical solutions to Lagrange’s energy

equations (Lagrange, 1778, 1973; Panza, 2003) rather than the

crucial role played by collisions and geometric motion in

Lazare Carnot’s algebraic mechanics or Faraday’s experimental

science (Faraday, 1839-1855; Heilbron; Pisano, 2013) with

respect to André-Marie Ampère (1775-1836) mechanical ap-

proach in the electric current domain and finally the physic

mathematics choices in James Clerk Maxwell (1831-1879) elec-

tromagnetic theory (Maxwell, 1873; Pisano, 2013). Physical

science makes use of experimental apparatuses to observe and

measure physical magnitudes. During and after an experiment,

this apparatus may be illustrated and/or and designed. Gener-

ally, this procedure is not employed in pure mathematical stud-

ies. Thus, one can claim that experiments and their illustrations

can be strictly characterized by physical principles and magni-

tudes to be measured. A modelling of results of the experiment -

tal apparatus allows for the broadening of the hypotheses and

the establishment of certain theses. If one avoids study-model-

ling experimental results, one may generate an analytical scien-

tific theory since there is no interest in the nature of physical

magnitudes and their measurements.

In the Descartesian rules on the collisions this eventual lack

mostly concerns: a) Descartes, in physics, guessed the impor-

tance of conservation principles, but, in the collision rules, he

was not able to exploit this fundamental and correct idea in a

suitable manner; b) a lack of an adequate mathematical inter-

pretation which could be helpful for the fully comprehension of

a phenomenon; and c) the impossibility to operate adequate

measure since the lack of fully knowledge of the concept of

physical quantities for some substances (i.e., one can think of

the concept of velocity, rather mass, or temperature, heat etc.)

25“Nec ista egent probatione, quia per se manifesta”. (Descartes, 1897-1913,

[Principia, VIII]: p. 70, line 12). The translation is ours.

26On that Thomas Samuel Kuhn (1922-1996) proposed (Kuhn, [1962] 1970)

that some contradictions between facts and theory are simply ignored by the

scientists until a dominant paradigm provides exhaustive explanations o

the majority of phenomena in which, in a certain period, the scientific

community is interested. In particular see Anomaly and the Emergence o

Scientific Discoveries (Kuhn, [1962] 1970, Chap. VI: pp. 52-65) and the

The Response to Crisis (Kuhn, [1962] 1970, Chap. VIII: pp. 77-91, in par-

ticular pp. 80-82; see also Osler, 2000). Besides that, Paul Feyerabend

(1924-1994) has shown—

asically through an analysis of Galileo’s

work—that some experiences are often neglected and that the critics o

experience is constituted by ad hoc argumen tation s ideated by the s cientis ts

to a c hi e v e h i s /h e r th e o re t i ca l p ur p oses (Feyerabend, 1975, chaps. 5-8; see also

199 1 ) . On G al il eo , r ece n tl y o n e c an s ee : F esta, 1995, Pisano, 2009a, 2009b,

27One of us stressed the relationship between physics and mathematics in

the history of science by means of many studies. Among physicists,

mathematicians, historians and philosophers who are credited with study o

mathematical physical quantities by means of experiments, modelling,

roperti es, existences, stru ctures etc. on e can strictly fo cus on how ph ysics

and mathematics work in a unique discipline physics mathematics (or, i

one prefers, mathematics physics). Thus, it is not a mathematical applica-

tion in physics and vice-versa but rather a new (for example in the 19th

century) way to consider this science: a new discipline physics mathematics

and not mathematical physics, where the change in the kind of infinity in

mathematics produces a change in both significant physical processes and

interpretations of physical quantities (Pisano, 2013: pp. 39-58; see also

2011: pp. 457-472).

28For instance, one can see an analogous situation concerning heat and

temperatu re concepts in the an alytical theory of heat (Fourier,1807, 1822)

with respect to Sadi Carnot’s thermodynamic theory (Pisano, 2010, 2011;

Gillispie & Pisano, 2013; Pisano, 2010). I briefly note that physics consid-

ers the indispensable agreement between theoretical data and observa-

tions—experimental data (including the properties of magnitudes) to estab-

lish a physical theor y. Generally, such arg uments are not considered rigo r-

ous by physics mathematics.

P. BUSSOTTI, R. PISANO

114

and the lack of experiments on collision; even considering the

whole described picture, it is hard to conceive how it is possible.

Thus, a conclusion would be that, in many cases, the lack of a

mathematical quantitative treatment prevented Descartes to

realize his procedures were not always correct.

Quantification of Physical Reasonings

In his physical works Descartes frequently takes position

against the essentialism typical of the Scholastic because he

thinks that such an approach cannot help in anyway to under-

stand the physical phenomena. Particularly he points out the

necessity to give a clear and quantitative form to the principles

and to the problems themselves of physics. However, Des-

cartes’ physical conception is not free from essentialist aspects.

For example

[...] a consequence of his first law of motion, Descartes

insists that the quantity conserved in collisions equals the

combined sum of the products of size and speed of each

impacting body. Although a difficult concept, the “size”

of a body roughly corresponds to its volume, with surface

area playing an indirect role as well. This conserved

quantity, which Descartes refers to indiscriminately as

“motion” or “quantity of motion”, is historically signifi-

cant in that it marks one of the first attempts to locate an

invariant or unchanging feature of bodily interactions29.

Descartes seemed to have understood—in the second dis-

cours of the La Dioptrique on the decomposition of the motion

and of the determination of a motion—that velocity has a direc-

tion30 besides a modulus (Descartes, 1897-1913, VI: pp. 93-

105). While, in the Principia no quantitative specifications with

regard to the role that the modulus and the direction31 of veloc-

ity should get in the physical phenomena is provided.

In the end, concerning his quantitative physical reasonings

we can mai nly claim:

1) Important concepts and rules—as quantity of motion, and

significant rules, the seven collision laws—are introduced by

Descartes (Principia) in a manner that they could not be ex-

pressed in an adequate mathematical terms.

2) The three physical parts of the Principia (the second, the

third and the fourth ones) are inscribed into a physical conception

that, in many regards, is still linked to the Scholastic ontologism.

Relying upon his researches on the fluid dynamics exposed

in the second book of his Principia, Newton argued that the

vortices, of which Descartes imagined the universe was com-

posed, cannot be stable (Newton, [1713] 1729, II, General

Scholium: pp. 387-388). This means the physics of Descartes’

Principia does not satisfy to the minimal requests for it to be

translated into quantitative terms in a quantitative model. If

Descartes had tried this operation, likely, he would have no-

ticed the physical and factual inconsistencies to which his prin-

ciples brought. For example, he could have seen that his colli-

sion rules were self-contradictory. Coherently with what he

himself had claimed in various passages of Regulae ad direc-

tionem ingenii (Descartes, 1897-1913, X: pp. 351-488; see Fig-

ure 11) and of Discours de la méthode (Descartes, 1897-1913,

VI: pp. 1-78). As a matter of fact, in the 13th regula Descartes

claims (Descartes, 1897-1913, X: pp. 430-438) that every prob-

lem has to be divided and analysed into a series of enumerated

parts whose knowledge is absolutely certain.

According Descartes, for a phenomenon33, one should carry

out a large series of experiments and after a profound analysis,

to consider only those who are really suitable to comprehend

such a phenomenon and to exclude the others. Moreover, this

reasoning is also valid as to single parts of an experiment. Thus

Descartes should have specified the experiments on which his

collision rules were based. This would have been important to

interpret into mathematical terms their results and in order to

make sure the concepts which he used were perspicuous. Nev-

ertheless, as above seen, he acted in a completely different way

as to the collision rules (this was also the case for concepts as

force, pressure, power). Thus, he would have realized how

difficult a satisfactory introduction of a conceptual structure

suitable to explain the physical phenomena is. In other words,

[Descartes] expresses so well the basic idea of a mathe-

matical physics, but he fails to specify how he want to

make sure physics susceptible to mathematical treatment.

For sure, he completely underestimated the difficulty of

this task: it is clear when with a candor typically of scho-

lastic he propose as example to represent by means of a

Figure 11.

Regulae ad directionem ingenii32.

29Slowik, § 4 . Author’s quotations.

30This does not mean that h e had completely caught t h e conce pt o f vector.

evertheless his conception might be indented as orientation ( directionand

versus).

31This happens i.e. more than once, at the beginning of the second discours

of the La Dioptrique. The word recurs by Descar tes to indicate a direction

is “costé” (i.e. see: Descartes 1897-1913, [La Dioptri que], VI: pp. 94-95).

32Descartes, 1897-1913, X: p. 430 [pp . 351-488]).

33Descartes, particularly quotes Gilbert’s experiments with the magnets

(Descartes, 1897-1913, X : pp. 430-438 ).

P. BUSSOTTI, R. PISANO

line the degree of whiteness, without making allusion to

the difficulties inherent in the measurement of a qualita-

tive intensity34.

In the Principia, an eloquent mechanistic conception (Dijks-

terhuis 1961) is provided: “[...] every modification of the matter

as well as the diversity of all its forms depends on motion

[...]”35. The interplay between gradation and intensity of the

considered quantities plays an essential role especially when he

enumerates some of the fundamental properties of the parti-

cles composing the three elements (Descartes, 1897-1913, X,

III part, § 52 and § 53: pp. 105-107). The intensity of the veloc-

ity of these particles decreases with a continuous gradation

from the first to the third element: the first one is composed of

very small fluted particles whose motion is extremely quick;

the second one by small circular particles, that anyway are a

little bit bigger and less quick of those composing the first ele-

ment; finally, the third element is constituted by the biggest and

slowest particles. The form of the particles and the different

intensity of their speed is the cause of being the first element

the luminous, the second the transparent and the third the

opaque. Given these presuppositions, one could expect a tran-

scription of all these physical relations into quantitative terms,

also considering that many scientists (Galileo is the most fa-

mous example) had already given a mathematical form to their

physics. Actually, in the Principia there is no mathematization

of the physical relations (Panza, 2006).

On the Two Essays, La Dioptrique

and Les Météores

The two Essays were written as appendices to the Discours

de la méthode to illustrate concrete applications of the theoreti-

cal precepts previously exposed. However, La Dioptrique and

the Les Météores , do not give the impression to follow pre-

established methodological precepts (Braunstein), rather they

show the lively work of the scientist and because of this they

are so interesting. The language used by Descartes is the French

because these texts also had a practical scope (construction of

lenses and telescopes) and therefore they had to be understood

by the artisans who, in general, were not confident with the

Latin. The purposes of the La Dioptrique and of the Les Mé-

téores are clear. It is possible to identify four conceptual centres,

whose treatment is based on rather diversified methodological

approaches (see Figure 12):

Since the second conceptual centre is the most significant

from a historical-scientific point of view and it is the one in

which Descartes follows explicitly a quantitative approach, we

will address two themes treated there: 1) the law of refraction; 2)

the rainbow. We will see that the approach is completely dif-

ferent from that connoting the Principia.

Reflexions on the Law of Refraction

In the second discours of the La Dioptrique Descartes (see

Figures 13(a) and (b)) determines the law of refraction.

The main purpose of the Dioptrique was the improvement of

optical instruments. To this end, Descartes derived the sine law

of refraction by analogy with the inflection of the motion of a

tennis ball upon entering water40.

Let us imagine (Figures 13(a) and (b)) that a ball K is

thrown from A to B and that it meets the surface of the cloth

CBE in B. If one supposes that K has a sufficient power to

break the cloth, then the ball will continue its movement be-

yond the cloth, losing a certain fraction of its velocity—let us

suppose half of the initial velocity—Descartes claims that, if

the determination (Descartes, 1897-1913, VI: p. 97, line 14 and

following pages) of the movement is decomposed into two

components, the one parallel and the other perpendicular to the

cloth, only the perpendicular component will be modified by

the encounter with the cloth, while the parallel will not be. If

now the three perpendiculars AC, HB, FE to CBE are drawn, so

that HF = 2AH, the ball will reach the point I of the circumfer-

ence of radius AB in a time which is the double to the one

needing to cover the part AB. A questions arises: how is it pos-

sible to determine the point I? The ball maintains its determina-

tion to proceed in the horizontal direction, therefore it will

cover a double space in a double time in the direction parallel to

the cloth. Thus, the point I is the one whose projection BE on

the cloth CBE is the double of CB. Descartes argues that if a

means is posed at the place of the cloth, that, as the water, op-

poses a major resistance to the motion of the ball than the air

(supposed to be over the water), then the law determining the

change of the ball motion in the passage from one mean to the

other one, is the same as the law determining the passage of the

ball between two portions of the same means separated by the

cloth.

Let us now suppose (Figure 13(b)) that the ball, once

reached B (let t be the time needed for the passage from A to B)

does not miss its velocity41, but rather receives a push so that

the velocity increases by 1/3. In this case, if we carry out a

construction analogous to the previous one, the ball will reach

the point I in a time equal to 2/3t, so that the projection BE on

the separating surface is equal to 2/3CB. This depends on being

the velocity along the horizontal determination unmodified.

Descartes claims that the action of light has the same behaviour

as the motion of the ball (Buchwald). Hence, if a light ray starts

from a less refracting means and reaches a more refracting

means (according to Descartes, this happens, for example, in

the passage from the air to the water) the component of motion

determination parallel to the separating surface will remain

unmodified, whereas, according to the nature of the two means,

the perpendicular component will be modified. Therefore as in

the following Figure 14:

The ratio between segments as KM and NL is invariant,

namely

...

KM AH

NL gI



34Dijksterhuis, [1950] 1977: p. 71, see also pp. 60-82. The translation is

ours. Still interesting i s Dijksterhuis, 1961.

35Descartes, 1897-1913, VIII: pp. 52-53. The translation i s ou rs.

40Darrigol 2012, Chap. 2. (Author’s italic). For our aim, in this highlightly

book an interesting account concerning Newton’s optic is presented (Ivi,

Chap. 3).

41In this context Descartes often speaks of force de son mouvement (a kind

of motion force: Descartes, 1897-1913 VI: p. 100, lines 1-2) and also uses

the word vitesse (velocity). In our specific case, the translation of

orce de

movement with velocity does not look to betray Descartes’ thought.

orce

de movement looks a conceptsimilar to quantity of motion, but since the

mass is an invariant in the interaction described by Descartes, the transla-

tion velocity looks appropriate. In this case, too, the lack of scientific con-

cepts and of a language universally codified makes these notions similar to

various post-Newtonian concepts in the history, but not perfectly identifi-

able with them.

P. BUSSOTTI, R. PISANO

116

1) The first discours of the La Dioptrique poses, in substance, the

bases for the prosecution of the treatment. Descartes carries out some

considerations on the nature of the light (in part they are used and

developed in the successive discours), but specifies that in the Essays

he will deal with the problems how the light is spread rather than what

is its nature36. Given these aims, Descartes explains he will limit to

illustrate the easiest manner to conceive the light in relation to the

phenomena he has to clarify. He will rely upon experiences and hy-

potheses, as the astronomers do in order to describe the motions in the

skies. Therefore the first discours of the La Dioptrique represents the

true methodological introduction to the two “physical” Essays rather

than the Discours de la méthode; 2) The second conceptual core,

which is the broadest and the most important, includes the discours

II-IX of the La Dioptrique and the discours VIII-X of the Les

Météores. Here Descartes faces, in the La Dioptrique, the theme of

the refraction, of the form of the eye and of the vision-mechanism, of

the properties of the lenses and of the most suitable form the lenses

must have to reach their purpose (correction of sigh-defects, magnifi-

cation of the objects, and so on). In the discours VIII, IX and X of the

Les Météores Descartes exposes the theory of the rainbow and of the

parhelions. The treatise is developed in a quantitative form. The au-

thor resorts to the experiments and reaches to explanations of phe-

nomena that, even if not correct or based upon correct presupposi-

tions, provide anyway a substantially perspicuous picture of the phe-

nomena; 3) The third conceptual core includes the 10th discours of the

La Dioptrique. This core could be defined the practical one because

Descartes proposes projects of machineries to construct optical in-

struments with as most perfect as possible lenses. This is an interest-

ing document of history of scientific technology (even if almost no

one of the projected machines was built); 4) Finally, the initial seven

discours of the Les Météores, that concern subjects as the nature of

the winds, the clouds formation, the causes of the precipitations, etc.,

have—in comparison to the other parts of the Essays—a style which

is nearer to the one used by Descartes in his Principia. Actually, sub-

jects dealt with in an original manner are not missing, as it is the case

in the sixth discours with regard to the form assumed by the snow-

flakes.

Figure 12.

The mechanism of the vision according to Descartes37.

Figure 13.

(a) Analogy between the movement of a bal l a n d a li g ht ray38; (b) The ball in B receives a push. Analysis of the consequences39.

36The referen ce text as to Descart esian id eas on li ght is Le Monde ou le Traité de la Lumière, published posthumous in 1664. The chapters 13 and 14 are those

specifically dedicated to light. Numerous paragraphs of the third part of Principia concern the nature of light inside the context of Descartes’ theory of matter

(Descartes, 1897-1913, VIII) Descartes tries to explain what light is, what its effects are, how the stars irradiate. The literature on Descar t es i an theor y o f l i g ht is

huge and it is impossible to provide even general indications. Recently for a very good and definitive history of optic see Olivier Darrigol (Darrigol,2012, and

all referen ces cited). The Essays co nstitute the second volume of the Italian translatio n of Descartes’ scientific works (Descartes, 1983). The editor Lojacono

added an adequate list of references and suggesting (Descartes, 1983: pp. 95-110; see also: Sabra, 1967; Tiemersma, 1988; Malet, 1990; Armogathe, 2000;

Schuster, 2 000; Shapiro, 1974; Schuster, 2013).

37Descartes, 1897-1913 [La Dioptrique, V discours], VI; p. 116.

38Descartes, 1897-1913 [La Dioptrique, I discours] VI; p. 91.

39Descartes, 1897-1913 [La Dioptrique, II discours] VI; p. 1 00.

P. BUSSOTTI, R. PISANO

Figure 14.

Determination of the motion and it s components42.

These previous segments are the sinus of the incidence and

of the refraction angles respectively. The law of refraction is,

thus, formulated like this: the ratio between the sinus of the two

angles is a constant and depends on the refraction index of the

two means (Descartes, 1897-1913, VI: pp. 101-102).

The argumentation proposed by Descartes does not claim to

be a demonstration of the refraction law, rather an explanation

that makes it deductively plausible. Nevertheless, there are

many questions concerning the picture proposed by Descartes:

1) What is exactly the determination of a motion?

2) Can the analogy of the ball that perforates the cloth (see

Figure 13) be legitimately extended to the light rays?

3) Does the nature of light have an influence on the refrac-

tion law?

These questions are, as a matter of fact, doubts on the legiti-

macy of Descartesian argumentation. On the other hand, New-

ton represents a conceptual and linguistic line of separation for

physics because the concepts used in pre-Newtonian physics

were—in general—not defined. One holds on the common use

of the words, or one oscillates between the common use and

forms of specification that were not always univocal or even

mutually coherent43. Therefore no surprise if Descartes did not

define the concept of determination. In any case, the determi-

nation does not look tout court identifiable with the direction of

a movement because Descartes uses the word direction, too,

when he speaks of movement. The concept of determination

has been studied for a long time by Descartes’ scholars (many

in: Mclaughlin, 2000) because undoubtedly it is difficult to

enucleate. It is maybe possible to think that Descartes intended

by determination the tendency of a body to reach the points of

its actual direction. These point are really reached or would be

reached if no impediment subsists. In the example of the second

discours of the La Dioptrique, the ball has a determination

towards D, however this point is not reached because of the

impediment of the cloth. The determination would hence be a

tendencies inherent to the motion of the body, while the direc-

tion is a geometrical line. This is an interpretation because the

concept of determination remains, in any way, problematic.

As to the analogy between motion of the ball and action of

light, Descartes assumes it without any further discussion and

justification. He underlines anyway that this analogy is not

complete because the ball is deviated far from the normal to the

surface of the cloth by the cloth itself, while if a light ray passes

from a less dense to a more dens means, the ray approaches the

normal, as well known. This brought Descartes to the wrong

conclusion that, given two means with different density, light,

in its movement, encounters less resistance in the more dense

means. However, according to Descartes the light propagates

instantaneously in every means. Therefore one cannot claim

that, according to Descartes, the light speed is major in a dense

means rather than in a less dense means. Rather Descartes ex-

plains that since light is “[…] an action by a very subtle matter

that fulfils the pores of the other bodies”44. Such action is hin-

dered by more “soft” bodies, as air, rather than by less “soft”

bodies, as water. Hence, as light encounters less resistance to

spread in the water rather than in the air, this is the reason why

in the passage air/water light approaches the normal.

The conceptual equipment used by Descartes to determine

the refraction law is hence tied to notions that are not always

well defined (as the one of determination of a motion), to

analogies and to wrong ideas; despite this the formulation of the

law is correct45. This induces us to think that the whole equip-

ment exposed to the reader in the second discours of the Diop-

trique is not directly connected to the way in which Descartes

discovered the refraction law. Rather it looks to have the aim to

convince the reader and to frame optics inside the mechanistic

project Descartes had already in his mind when he wrote the

Essays. In a brilliant and profound paper Schuster underlines

that:

Descartes was willing to try to ride out likely accusations

that the premises are empirically implausible, dynamically

ad hoc, and in some interpretations, logically inconsistent,

because the premises provided elegant and more or less

convincing rationalisations for the geometrical moves in

his demonstration46.

The premises were confused and wrong, but the model was

elegant and worked. Schuster produces convincing evidences in

favour of the thesis that the refraction law was ideated by Des-

cartes through an itinerary based upon his studies of geometri-

cal optics. If this is true, the law was deduced independently of

dynamic considerations added by Descartes in a second time.

Conceptual Streams behind Descartes’ Law of

Refraction

The physical-geometrical core of Descartesian argumenta-

tion can hence be connected to the idea of decomposing a

42Descartes, 1897-1913 [La Dioptrique, II discours], VI: p. 101.

43In this sense, a classical example is the concept of force. Many scholars

used it in the 16th and 17th centuries. There is an abundant and interesting

literature on this subject, that allowed—in great part—to clarify how dif-

ferent authors used this term. In this case, too, before Newton had given his

definition of force, this w ord did not have a univocal meaning.

44“[…] une action reçue en une matière très subtile, qui remplit les pores

des autres corps […]” (Descartes, 1897-1913,VI: p. 103, lines 13-14). The

translation is ours.

45We do not enter here into either the problem concerning the relations

etween Descartes and the other authors who, substantially, had understood

refraction law, as Willebrord Snel van Royen called Snellius (1580-1626),

Claude Mydorge (1585-1647) and Thomas Harriot (1560-1621) or the

fundamental role Johannes Kepler (1571-1630) had in this studies on re-

fraction (Pisano and Bussotti 2012; see also Malet 1990). The notes to the

first and second discours of the Dioptrique (Descartes 1983) are thorough

in this regard. See also Schuster 200 0.

46Schuster, 2000:p. 271. Particularly Schuster has recently published an

important contribution to the mechanistic Descartes’ conception (Schuster

2013

)

P. BUSSOTTI, R. PISANO

118

movement in two mutually perpendicular components. Des-

cartes could hence imagine to decompose the motion of a light

ray into these two components, without introducing the me-

chanical analogy of the ball or the concept of determination.

His convictions on the nature of light induced him to introduce

these notions. There was no necessity connected to the physi-

cal-geometrical argumentation to do that because the argumen-

tation itself would have lost nothing of its validity without the

mechanical analogy of the ball.

The further element to take into account is that Descartes led

many experiments concerning the refraction and optics in gen-

eral. In order to prove this, three examples are indicative: a) at

the beginning of the third discours of the La Dioptrique, the

experiment on the way in which eye forms the imagines (Des-

cartes, 1897-1913, VI: pp. 105-106. The problem of the vision

is further specified in the fifth and sixth discours, of the same

work, pp. 114-147); b) in the tenth discours of the same text the

affirmation that, in order to establish the most suitable form for

a hyperbolic lens and the best position for its focus “[…] ex-

perience will teach better than my reasoning”47. Thus Descartes

realized that the “[…] exact proportions are not so necessary

that they cannot be changed a little bit”48. Hence, in this case,

geometry is a guide for the form of the lens, but it does not

determine such form in an absolute and univocal manner; c) the

experiments with an ampoule full of water (VIII discours of the

Les Météores) to comprehend the rainbow phenomenon. The

experiments in optics are hence a fundamental aspect of Des-

cartes’ works. The genesis of the discovery of refraction law

can perhaps summarized this way:

1) Descartes knew the tradition of geometrical optics studies;

2) He had realized—and this is a great, even if not exclusive,

merit of his—that the physical phenomena can be understood

only if quantities that remain invariable are determined;

3) He had carried out a plurality of experiments. All these

facts brought him to intuit and to formulate the refraction law in

a correct way. The other argumentations we have seen, were

introduced because of philosophical convictions and to make

the law plausible, but they do not play a role in the discovery of

the law and—it is necessary to add—they are extraneous to the

nature of the phenomenon.

The situation for the case here analysed is far different from

that of the Principia: the refraction law is a paradigmatic ex-

ample of a reasoning in which the mathematical apparatus is

poor, but an easy formalization of Descartes’ reasoning shows

its consistency and correctness. In fact, it is enough:

1) To decompose the motion of light in a vectorial form49,

according to the parallel and perpendicular components to the

incidence surfa ce.

2) To use a symbolic notation to indicate the angles.

3) To introduce the concept of incidence and refraction angle.

All this is clear in Descartes’ treatise, even if the reasoning is

not completely symbolized. Therefore a coherent quantification

is possible, while it was not the case with the collision rules

introduced in the Principia.

The La Dioptrique shows that a mathematical structure exists

at the basis of Descartesian reasoning: let us consider the VIII

discours, where Descartes exposes the focal properties of the

parabolic and hyperbolic lenses. Furthermore, in the last part of

the second book of his Geometry, Descartes extends the study

of reflection and refraction to the oval lenses (Descartes, 1897-

1913, VI: pp. 424-441). The treatment is, in this case, highly

formalized and completely expressed in mathematical terms.

Descartes carried out experiences and experiments. He mathe-

matized the results and proposed explicative models based on

the quantification of the phenomena. Because of this, the role of

Descartes in the scientific turning point of the 17th century is

relevant. The concepts of gradation and intensity represent an

interesting instrument through which Descartes’ ideas on re-

fraction can be interpreted. First of all, gradation is the basis of

refraction itself: if the different transparent materials had not

different refraction indices, the phenomenon itself would not

exist. Therefore gradation of the refraction indices represents

the basis of this optical phenomenon. Since every material has

its own index, it is possible to construct a graduated scale: the

refraction indices represent the intensity with which every ma-

terial refracts light. Descartes, by discovering the exact form of

refraction law, made the intuitive idea that the materials have

different refraction powers perspicuous. In this manner he ide-

ally established a scale of gradation, even thought Descartes

ideas that most dense materials also are the most refracting is

wrong.

The Rainbow

The eight and most important discours of the Les Météores is

dedicated to the rainbow (Maitte, 1981, 2006; Ronchi & Ar-

mogathe, 2000).

The way in which Descartes faces the rainbow problem pre-

sents an excellent epistemological model for the genesis of the

scientific discovery. It is also indicative of the non univocal

manner in which Descartes addressed the problems of physical

arguments. The most significant aspects are three:

1) The use of experience to catch the properties of the phe-

nomena;

2) Scientific quantification of the reasoning to obtain per-

spicuous results;

3) Elements of Descartes’ mechanistic conceptions that in-

fluenced his rainbow theory.

From the beginning, Descartes resorts to experience in an

appropriate manner (Descartes, 1897-1913, [VIII discours], VI:

pp. 325-327). In an initial phase of his work, his purpose is to

realize, in a qualitative manner, what the invariants character-

izing the rainbow phenomenon are: since rainbow is visible not

only in the sky, but also, for example, in the fountains in which

the water is illuminated by sun rays under particular conditions

depending on the way in which the sun rays hit the drops of

water in respect to the observer, Descartes reasoned this way

(see Figure 15, on the right our paraphrase).

This means the rainbow is not necessarily connected to at-

mospheric events as the rains. Furthermore Descartes observes

that the size of the water drops does not have any influence on

the phenomenon. He remarks that if the ampoule is raised and

suspended by a machinery, which is not described in the text,

but that can be easily imagined, the conclusion is the following:

a sun ray hits the ampoule (that is the drop of water) in B, it is

refracted by the water in C. From here it is reflected in D, from

47“[…] l’experience enseignera mieux que mes raisons” (Descartes,

1897-1913, VI: p. 20 2, lines 9-10). The translation is ours.

48“[…] proportions ne sont pas absolument nécessaires, qu’elles ne puissent

beaucoup être changées”. (Descartes, 1897-1913, VI: pp. 201-202). The

translation is ours.

49Even if, probably, Descartes did not catch the concept of vector in its

generality, he himself proposed to decompose the determination of a mo-

tion into two m utually perpendicular components, as we have seen.

P. BUSSOTTI, R. PISANO

[…] this arc [the rainbow] can appear not only in the sky, but also

in the air near us, every time there are many drops of water illumi-

nated by the sun, as the experience shows in some fountains.

Therefore I established easily that the rainbow depends only on the

way in which the light rays act on the drops and on the inclination

with which the rays reach our eyes from the drops50. If AB or ZM

indicate the direction Sun-eye, when the angle DEM is about 42

degrees, t hen a brillian t red appear s in the par t D. Such colour con-

tinues to be present whatever is the movement of the ampoule, as

long as th e angl e DEM remains 42 degrees. As soon as this angle is

increased, even of a very small quantity, the red disappears. While,

if the size of the angle is reduced, the red pencil of light is divided

into less brilliant pencils in which the other colours of the rainbow

appear. If the size is further diminished, every colour disappears.

However, when the angle KEM is 52 degrees, the zone K is illumi-

nated by a red, that is less brilliant than the one present in D when

EM is 42 degrees. If the angle KEM is made broader, the other

colours appear in zones as Y. These colours have a minor intensity

than the red in K. If the size of the angle is either slightly dimin-

ished or made it much bigger, every colour disappear. It is likely

that at this stage, Descartes had already understood the role of re-

flection and refraction in rainbow genesis. However, to have a con-

firmation he carries out the following experiment: he poses an ob-

scure and opaque body in one of the points of the lines AB, BC, CD

e DE. He remarks that the red colour disappears. While, if the

whole ampoule is covered, excluded the points A, B e D, an d no ob -

stacle disturbs the action of the rays ABCDE, the red continues to

be present.

Figure 15.

Explanation of the r ainbow in the Les Météores51.

where it is refracted to the observer in E. Consequently the red

appearing in D is given by two refractions and one reflection.

The red in K of the second rainbow is given by one refraction

of the ray in G, followed by one reflection in H, a further re-

flection in I and a refraction in K until the ray reaches E. Since

there are two reflections and two refractions, the red is less

intense. In this way, the general nature of the phenomenon is

explained. Two questions are still to be answered:

1) Why does the rainbow appear when the angles DEM and

KEM are respectively 42 and 52 degrees?

2) What is the cause of the rainbow colours?

Descartes answers the first question through the following

very acute reasoning: let the drop of water be represented by

the circumference (see Figure 16).

In the picture traced by Descartes there are many elements

that characterize a great part of the scientific discoveries:

1) The use of the experience to achieve a global qualitative

vision of the studied phenomenon.

2) The resort to the experiment having in mind not only the

questions, but also a series of possible answers.

3) The quantification of the data and resort to the demonstra-

tion to explain the phenomena in a perspicuous way.

The concepts of gradation and intensity represent once again

Let F be the point of the drop in which the solar ray strikes. Let this ray

be refract ed in K, from K r eflect ed to N and from here refracted towards

the eye in P or ref lect ed t o Q and from Q refract ed t owar ds t he eye i n R.

This figure is hence a model of the first and of the second rainbow.

Traced the perpendicular CI to FK from the centre C of the circumfer-

ence, it results that HF

CI is the ratio between the refraction indices of

the air and of the water. In fact, let us trace the radius CF and the tan-

gent at t h e ci r cu mf er en ce i n F, in the triangle FCC', the sine of the angle

C'FC is CC' = FH, but C'FC is equal at the incidence angle and CFI is

the angle of refraction. Since the refraction index water-air is known,

the ratio of FH to the radius CD is kn own. It i s therefo re poss ible to de-

termine the ratio of IC with these two quantities. Thus, it is possible to

establish the size of the arcs FG and FK hence to calculate the angle

ONP. If the position of the point F in which the solar ray strikes the

drop varies, the angle ONP will vary in a way that can be calculated.

The calculation shows that, when F varies—it is enough to limit the

analysis at the quarter of circumference AD—the rays that come out

with an ang l e ONP of about 40 degrees are more numerous than the rays

that come out with other angles. This explains why the first rainbow is

visible when the angle DEM in figure 9 is about 42 degrees. In an

analogous way, it is possible to prove that most part of the angles SQR

are about 52 degrees. This explains th e second rainbow.

Figure 16.

Geometrical model of the rainbow in the Les Météores52.

50“[...] cet arc ne peut pas seulement paroistre dans le ciel, mais aussy en

l’air proche de nous, toutes fois & quantes qu’il s’y trouve plusieurs gouttes

d’eau esclairées par le soleil, ainsi que l’expérience fait voir en quelques

fontaines, il m’a esté aysé de iuger qu’il ne procède que de la façon que les

rayons de la lumière agissent contre ces gouttes, & de là tendent vers nos

yeux”. (Descartes, 1897-1913 [VIII discours], VI: p. 325, line 10). The

translation is ours.

51Descartes, 1897-1913 [VIII discours], VI: p. 326. 52Descartes,1897-1913 [VIIIdiscours], VI:

. 337.

P. BUSSOTTI, R. PISANO

120

a lens through which the physical phenomenon can be observed

and the work by Descartes interpreted: gradation and intensity

are inherent (Figure 15) to the angles between the lines EM,

ED and EM, EK. When the degrees of these two angles vary,

the two arcs of the rainbow either subsist with the different

colours or disappear at all. The intensity of the colours is a

function of these angles, too. The Figure 16 can be interpreted

as a model that provides a geometrical representation of the

gradation of the angles. The reasoning and the calculation

demonstrate why those particular sizes (42 and 52 degrees) are

the critical ones for the rainbow. Inside a context that, for many

aspects can be defined of a modern physical context, the angu-

lar gradation justifies the existence of the rainbow. This grada-

tion is connected to the intensity of chromatic gradation by an

elegant functional link.

For sure Descartes aims to explain the nature of the colours.

Thus, he provides an interesting answer, in part based on ex-

periments carried out with an optical prism and in part on his

mechanistic convictions. Namely, in his theory of the colours,

he takes into account the empirical data, but tries to explain

them by means of presuppositions tied to the way in which he

conceives the nature of light. If, as to refraction law, the anal-

ogy of the balls was not the central core of the argumentation,

here the idea that the action of light is transmitted by the parti-

cles of the subtle matter is essential for the explanation.

Descartes constructs a mechanic model in which he imagines

that a subtle matter composed of little spheres having a deter-

mined velocity of translation is present. The reciprocal colli-

sions among these particles and/or the collisions with some

other body can modify this velocity and also induce a rotational

motion in each single particle. He claims the colours depend on

the motions of the particles that constitute the subtle matter and

that transmit the action of light. At different velocities of rota-

tion correspond different colours. This idea is brilliant, but

(differently from what had happened in the rest of the discours

on the rainbow) no relation between modelling and physical

phenomenon is shown; such relation is only supposed. Fur-

thermore a direct connection between the experiment with

prism and the supposed explanation of this experimental result

is missing. There is no demonstration. Thus Descartes replaced

the facts of the chromatic world with a set of other facts relative

to the motion of the supposed particles, but these motions are as

difficult to be explained as the colours themselves. The model-

ling proposed is not easier than the phenomenon because it

contains the same number of elements: simply Descartes re-

places the facts of a certain world with the facts of another

world. There is no precise assumption that explains why the

world of the particles is, from an epistemological and physical

point of view, easier than the chromatic world and justifies

hence why the world of the particles should provide an expla-

nation for the chromatic world. Because of this, the model is

not explicative. In a se nse, the reasoning could be inverted until

reaching a supposed explanation of the motion of the particles

by means of the colours and not vice versa. The way of reason-

ing proposed by Descartes in this case is more similar to that of

the Principia than to the one connoting the rest of the discours

on the rainbow, even if the form in which the subject is exposed

and the fascination of Descartesian speculations give an ap-

pearance of plausibility to Descartes’ theory of colours. Again,

in this case, gradation and intensity are also cardinal concepts.

When the gradation of the velocities and of the motions of the

spheres of the subtle matter vary, a corresponding variation of

the chromatic scale exists. Therefore the two concepts of gra-

dation and intensity can provide a good perspective description

through which to analyse Descartes’ physical works.

Conclusion

Notes on Science & Society Civilization

Usually a discussion concerning history of science and tech-

nique/technology is presented such as a discipline within the

history of science for understanding eventual relationship be-

tween science and the development of art crafts produced by

non–recognized scientists in a certain historical time. The rela-

tionship between science and science and society and conse-

quent civilizing by science is centred on the possibility that the

society effetely developed a fundamental organization in capac-

ity to absorb science and produce technologies (i.e., water and

electrical supply, transportation systems etc.). Thus, a devel-

opment civilization was necessary parallel to development of

the science within society? Is effetely happened that? Did Des-

cartesian and Newtonian physical works develop as a response

to the needs of society? Alexandre Koyré (1892-1964) strongly

remarked the history of science and the role played by mathe-

matics between Newton and Descartes (Koyré, 1965, Chapter

III) in the history of scientific thought. Through the intuition

that the fundamentals of scientific theories contain two basic

choices, Koyré’ intellectual matrix (Pisano & Gaudiello, 2009,

200b) has been cleared up.

The new science, we are told sometimes, is the science of

craftsman and engineer, of the working, enterprising and

calculating tradesman, in fact, the science of rising bour-

geois classes of modern society. There is certainly some

truth in this descriptions and explanations […]. I do not

see what the scientia activa has ever had to do with the

development of the calculus, nor the rise of the bourgeoi-

sie with that of the Copernican, or Keplerian, astronomy

theories. […] I am convinced that the rise and the growth

of experimental science is not the source but, on the con-

trary, the result of the new theoretical, that is, the new

metaphysical approach to nature that forms the content of

the scientific revolution of the seventeenth century, a

content which we have to understand before we can at-

tempt an explanation (whatever this may be) of its his-

torical occurrence53.

[...] I shall therefore characterize this revolution [the birth

of the modern science] by two closely connected and even

complementary features: (a) the destruction of the cosmos

and therefore the disappearance from science—at least in

principle, if not always in fact—of all considerations

based on this concept, and (b) the geometrization of space,

that is, the substitution of the homogeneous and abstract—

however now considered as real—dimension space of the

Euclidean geometry for the concrete and differentiated

place-continuum of pre-Galilean Physics and Astron-

omy54.

According to the Russian historian55 we can consider that: 1)

the history of scientific thought has never been entirely sepa-

rated by philosophical thought. 2) the most important scientific

revolutions have always been determined by a replacement of

53Koyré,1965: pp.5-6.

P. BUSSOTTI, R. PISANO

philosophical speculations. Thus i.e., the history of scientific

thought (i.e. for physical Descartesian and Newtonian sciences)

has not developed by vacuum, but it moves in a set of ideas,

foundational principles, or axiomatic evidences.

Final Remarks on Descartes’ Physical Works

Particularly in this paper we have highlighted the two differ-

ent ways in which 3) Descartes developed his physical re-

search-frames in his Physical Work s: a) typic al of an essential-

ist and aprioristic way of thinking, b) based on experiments

and mathematization. The differences between, the physical

Essays Dioptrique and Météores, and the Principia, concern

both the content and the methodological aspect. These differ-

ences regard the way in which the scientific work is addressed.

The approach of the Essays can be epistemologically inter-

preted:

1) Descartes presents his experimental and theoretical work

as a scientist.

2) He realizes that science and technique have deep connec-

tions. Therefore he had the idea to address the essays basically

to the artisans.

3) Descartes had close relations with export artisans as Fer-

rier. He fully understood the role that science could play in the

construction of machines.

4) Descartes thought machines were fundamental for the fu-

ture of mankind.

5) The mechanistic convictions of Descartes are apparent

from many passages of the two Essays, even if these concep-

tions do not play a fundamental role for the discoveries exposed

in the La Dioptrique and in the Les Météores.

The Essays present hence Descartes as a producer of the sci-

entific work. On the basis of his whole scientific work, likely

Descartes was one of the first scientists to have the idea that, in

a physical theory every phenomenon must be explained on the

basis of a precise law. However, the perspectives of the scien-

tific work were not completely rosy:

1) In Descartes’ epoch it was already clear that experiments

were the basis of physics and they would have been still more

in the future;

2) The costs for the research were increasing more and more.

From here the necessity of financial supports;

3) In Descartes’ time, ecclesiastical censorship continued to

represent a problem;

4) Descartes was profoundly surprised by Galileo’s con-

viction. He wrote in a letter to Mersenne in November

1633:

In fact, I cannot imagine that he, who is Italian and well-

liked to the Pope himself—as far as I know—was consid-

ered a criminal for no other reason but he wanted to estab-

lish the earth movement. I am aware that this conception

was censured by some cardinals. However—as far as I

remember—I had heard that it had continued to be taught

in Rome itsel f56.

Therefore in the first half of the XVII century the social and

political situation was difficult for the scientists. In fact, new

discoveries were emerging with a rhythm far more rapid than in

the previous centuries, but, at the same time, the financial sup-

ports to develop research depended on the power holders and

not on the scientists themselves. The aim of the power holders

was of course to use the science for their scopes. Furthermore

the ecclesiastic censorship was strong. If we take into account

this picture, it is perhaps possible to understand the approach of

the Principia. This book is the expression of personal Des-

cartes’ physical and metaphysical ideas (as the mechanistic

conception) and of his desire to become—despite his declara-

tion in a contrary sense—a sort of new scientific authority.

From here the encyclopaedic character of the book arises,

whose intention is to face all problems of physics. However, at

the same time, the Principia can be interpreted as a book pro-

foundly influenced by the social situation we have rapidly out-

lined. This situation is also connected to the most convenient

way to present science. Descartes wanted to make science ac-

ceptable to the Church and to the power people with whom he

would have had contacts. In this manner, for example, the

paragraphs III, 16-19 of the Principia can be explained, where

the validity of the systems of Ptolemy (II century AD), Nico-

laus Copernicus (1473-1543) and Tycho Brahe (1546-1601) is

denied. Descartes shows that, in his own system, the earth has

to be considered, as a matter of fact, at rest. This assertion can

be interpreted as an insurance for Catholic Church. The insis-

tence on the fact that, in determining physical laws, deduction

is far more important than experiments, is one of the most evi-

dent ideas expressed in the Principia. This idea is in part co-

herent with what Descartes really thought, but in part it has the

aim to show that experiment—namely a fine enquire on the

world that can modify the world itself—with its potentially

subversive value, has a secondary importance. In other terms:

science is not dangerous from a social point of view and can

hence be accepted by the power holders. As a matter of fact, we

have seen that, when Descartes produces science, he cannot

renounce to experiment. The Principia can hence been inter-

preted as the text that concludes a phase of the scientific revo-

lution, namely the phase in which the social role of the scien-

tists was not yet clear. The scientists were not yet, in every

aspect, institutional figures, as they became in the second half

of the 17th century57. Because of all these reasons, the Prin-

cipia are a text that presents a relevant historiographic interest.

It is not always easy to distinguish what Descartes wrote to

justify the scientific work and to make it acceptable to whom

could valuate this work potentially dangerous from a social

point of view from what he wrote for a real conviction. Fi-

nally, main comparisons between Descartes’s and Newton’s

conceptions we carried out summarized by the following

table as relevant dissimilarities between these two scientists,

with regard to the subjects deal with in our paper (see Table 1).

54Koyré, 1965: pp 6-7. In the following explaining of Alexandre Koyré’

choice for the history of science: “The destruction of the cosmos” that is a

replacement of the finite world, as it had been hierarchically classified by

Aristotle, with the infinite universe. “The geometrization of space”: that is a

replacement of Aristotle’ physical (concrete) space with the abstract space

of the Euclidean geom etry. (Pisano & Gaudiello, 2009a, 2009b).

55A conference (1954, Boston) of American Association for the Advance-

ment of Science. Cfr.: The scientific Monthly, 1955; Koyré, 19 71.

56“Car ie ne me suis pû imaginer, que luy qui est Italien, & mesme bien

voulu du Pape, ainsi que I’entens, ait pû estre criminalizé pour autre chose,

sinon qu’il aura fans doute voulu establir le mouvement de la Terre, lequel

ie sçay bien auoir elle autresfois censuré par quelques Cardinaux; |mais ie

pensois auoir oüy dire, que depuis on ne laissoit pas de l’einseigner pub-

liquement, mesme dans Rome” (Descartes, 1897-1913, I: p. 271, lines 2-9).

The translation is ours. Moreover, we remark that a possible pro

lem with

ecclesias tical cen so rsh ip in duced hi m to av oi d t he p ub licat ion o f Le Monde,

as he wrote in the same letter (Descartes, 1897-1913, I).

57By concerning the social and political situation in 17th century and the

relations with science, one can see the following works: Heilibron, 1979;

Dear, 1995, 1987; Kokow ski, 2004; Gorokhov, 2 011.

P. BUSSOTTI, R. PISANO

122

Table 1.

Descartes and Newton’s main arguing .

Descartes Newton

General Conceptions

A) In the Principia Philosophiae Descartes exposed his mechanistic

conception. An attempting to unify the whole physical world on

the bases of few principles and rule is present; the treatment was

not mathematized and accurate definitions are missing. The

geometric model proposed provided weak indications about the

positions of the bodies in function of the time. Important laws are

introduced as the

inertia law and the law of the conservation of movement. The main

reference are the Principia Philosophiae (Descartes, 1897-19 13,

Inertia law, VIII-1, II part, § XXXVII: pp. 62-63; the conservation

of movement: Ivi, § XXXIX and § XL: pp. 63-65). Nevertheless,

since a mathematical and definitional apparatus is missing, it is

difficult—and probably in part wrong—to interpret Descartesian

conceptio n o f inertia and quantity of

motion as in the Newtonian and p ost-Newtonian physics.

B) In the Essays, La Dioptrique and Les Methéores Descartes deals

with speci fic prob lems connect ed to r eflectio n and ref raction. Her e

the proposed models are mathematical or, at least, can be easily

acceptably mathematized; demonstrations are presented. The

refraction-law is expressed (Descartes, II Discours of the La

Dioptrique in: Descartes, 1897-1913, VI: pp. 93-105).

A) According to Newton, a mech anical mod el has to fores ee the posit ions

of the bodies through mathematical relations between the space

variable a n d t he time variable. In gene r al , a physical model must supply

precise laws and deductions expressible in a mathematical form (for

example in the Author’s Preface, Newton writes: [...] and then from

these forces, by other propositions, which are also mathematical, we

deduce the motions of the Planets, the Comets, the Moon and the Sea”

(Newton, [1713] 1729, I, Preface: p. A2; see also Ivi, II, General

Scholium: p. 392) Definitions (8), axioms or laws (3) are exposed in th e

initial section of the Principia (Newto n, [1713 ] 1729, I, Def initi ons: pp

1-18). Axioms or laws plus their corollaries: pp 19-40). For the first

time a physicist feels the need to provide definitions of the quantity he

is dealing with.

B) Generally speaking the structure of Principia looks Euclidean, but,

Newton introd uced his apparently abstract formulations in order to

explain, from a unitary point of view, physical p henomena.

Connection theory-experience

A) In the Principia Philosophiae there is an insufficient connection

between theory and experience. In some cases—as in the one of

the movements of the planets—there are intrinsic difficulties to

connect theory and experience because no provisional model is

supplied, but only a descriptive one. In other cases, as the collision

rules, experience is not coherent with theoretical rules (Descartes,

1897-1913, VIII-1, second part, §§ XLVI-LII: pp. 68-70).

Descartes ignore the experience, claiming that many conditions

can influence the experiments and the experience (Ivi, § LIII: p

70). But a critics of experience is lacking.

B) In the Essays the experience and the experiments play a

fundamental role. Descartes analyzed many empirical details and

explains them through the theory. The experience and the

experiments guided him to develop his theory (See i.e., how

Descartes focused on the experience of the rainbow colours to

explain this phenomenon in Les Météores (Descartes, 1897-1913,

VI: pp. 325-344).

A) The exp erien ce and the ex per iment s ar e th e bas es o f Ne wto n’s phys ics .

He is explicit in the Optiks, where a plurality of experiments are

presented and the theory is clearly constructed to provide a model to

the phenomena deriving from experiments (Newton, [1704] 1730: p.

1). A profound critics of experience is implicitly presented because

Newton specifies the experimental conditions and the effects that could

perturb the results of the experiments (Newton, [1713] 1729, II: pp.

202-205).

B) The Philosophiae Naturalis Principia Mathematica (Newton, [1713]

1729, et editions) too, have their sourc e of inspirat i ons in the

phenomena (no really experimental) and in the attempt to explain and

foresee the phenomena; i.e, the second section of the third book of the

Philosophiae Naturalis Principia Mathematica is titled The

Phaenomena or Ap pe ara nc e s (Newton, [1713] 1729, III: p p . 206-212)

Fundamental concepts: space, time, mass

A) In the Principia Philosophiae, the space cannot be distinguished

from res extensa. It is always relative (Descartes, 1897-1913,

VIII-I, II part, Question X: p. 45).The time is relative (Descartes,

1897-1913, VIII-I, I part, questions LVI-LVII: pp. 26-27).

B) It is known that a scientific (i.e., by magnitudes) distin ction

between mass and weight was problematic at that time.

A) The absolute time exists (before than in the famous general General

Scholium, Newton spoke of absolute space and time in the General

Scholium posed as a conclusion to his Definitions. (Newton, [1713]

1729, I: pp. 9-18). Both were introduced also as an answer to the

problems present in Descartes’ physics explained by Newton himself

(Newton, [1713] 1729, II, General Scholium: pp. 387-393).

B) The mass is clearly distinguished by the weight (Ivi, definition I: p. 1).

Even if Newton’s concept of mass can be criticized for well known

reasons, the concept of mass was (within his physical mathematical

system) reasonable at that time.

Problems connected to the cultural environment

A) Descartes had to face a series of problems that in Newton’s land

and time wee far less serious: 1) Catholic censure of Copernican

theory; 2) role of the scientist still not well defined. Power holders

could think that scientists were dangerous for the social order; 3)

scarce financial support. Since the Principia are Descartes’ world-

system and the most conspicuous manifest of his way of thinking,

he tried and intermediation between his ideas and possible

dangerous consequences.

B) This is one of the reasons why the Principia are such a tormented

text. In the Essay (La Dioptrique, Les Météores, La Géométrie)

dealing with more specific arguments, he did not h ave these

problems and the treatment was c l e arer and m or e coherent.

A) Newton did not deal with the problems addressed by Descartes because

of: 1) Different period: end of the 17th begin of the 18th century. 2)

Different land: England where the Catholic censure was not effective.

3) Newton represents a scientist who is perfectly integrated in the

system; rapidly the scientists were becoming persons with public roles

and well defined social positions.

B) Newton was completely free to publish his works without the problems

faced by Descartes.

C) Comment: Ne wt o n ’s t h eo r et i cal concepti on s ar e n ot dir ectly influenced

by the social and political environment. He was free, as to the subjects

he dealt with, and he had non problems with censure. The influence of

the social and political environment was indirect as far as it allowed

Newton to develop fre ely his rese arches.

P. BUSSOTTI, R. PISANO

When a thought does not have clear and delineated scopes, it

likely results tormented, often self-contradictory and difficult

to frame into an organic picture. Under this point of view, Le is

different from the Principia: in this original and pleasant brief

treatise Descartes exposes his mechanistic conceptions and his

theory of light, based upon them. But in this case, the scope is

clear—independently of the correctness of the basic ideas ex-

pressed in Le Monde—and the argumentation is linear. Finally,

even if, as Scott and Koyré claimed

Thus Descartes’ hypothesis at least has the merit of ex-

plaining the nature of weight without recourse to any oc-

cult force acting across space. More than that, it is easy to

detect in it a groping after a universal law; the mechanism

by which a body falls to the earth is in the last resort the

same as that which keeps the planets in the solar vortex

[…]58

[...] thought of course unsuccessful, attempt at a rational

cosmology, an identification of celestial and terrestrial

physics, and therefore the first appearance in skies of cen-

trifugal forces [...]59

Descartes had in physics the merit to have tried a unique ex-

planation for the gravity on the Earth and for the orbital move-

ments of the planets (unification of terrestrial and celestial

physics), this has happened with a form more coherent with a

mentality typical of an aprioristic conception of the physics

rather than the one connoting an observative-experiment-quan-

titative approach.

Acknowledgements

We want to express our gratitude to anonymous referees for

precious comments and helpful suggestions.

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