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
D
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
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
0
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
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),
L
a
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
f
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
f
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
f
two bodies upon each other are always equal, and directed to contrary
p
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
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
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-
p
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).
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
f
the majority of phenomena in which, in a certain period, the scientific
community is interested. In particular see Anomaly and the Emergence o
f
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-
(1924-1994) has shown—
b
asically through an analysis of Galileo’s
work—that some experiences are often neglected and that the critics o
f
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
f
mathematical physical quantities by means of experiments, modelling,
p
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
f
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
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
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.
29Slowik, § 4 . Author’s quotations.
30This does not mean that h e had completely caught t h e conce pt o f vector.
N
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-
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
f
orce de
movement with velocity does not look to betray Descartes’ thought.
F
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
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
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
b
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
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
ˆ
D
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:
p
. 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-
tion is connected to the intensity of chromatic gradation by an
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-
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
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
b
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
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