## 1. The Origins and Definition of Descartes' Method



The origins of Descartes' method are coeval with his initiation
into a radical form of natural philosophy based on the combination of
mechanics, physics, and mathematics, a combination Aristotle
proscribed and that remained more or less absent in the history of
science before the seventeenth century (on the relation between
mechanics, physics, and mathematics in medieval science, see Duhem
1905-1906, 1906-1913, 1913-1959; Maier
1949-1958; Clagett 1959; Crombie 1961; Sylla 1991; Laird and
Roux 2008). Descartes first learned how to combine these arts and
sciences from the Dutch scientist and polymath Isaac Beeckman
(1588-1637), whom he met in 1619 while stationed in Breda as a
soldier in the army of Prince Maurice of Nassau (see Rodis-Lewis 1998:
24-49 and Clarke 2006: 37-67). Beeckman described his form
of natural philosophy as "physico-mathematics" (see AT 10:
67-77 and Schuster 2013), and the two men discussed and
corresponded about problems in mathematics and natural philosophy,
including problems in the theory of music, hydrostatics, and the
dynamics of falling bodies (see AT 10: 46-47, 51-63,
67-74, 75-78, 89-141, 331-348; Shea 1991:
1-121; Damerow et al. 1992; Schuster 2013: 99-167).



While it is difficult to determine when Descartes composed his
principal methodological treatise*, Rules for the Direction of the
Mind (Regulae ad directionem ingenii*), it is widely believed that
he composed the *Rules* in the 1620s (see Weber 1964:
194-207; Gaukroger 1995: 104-187; Schuster 2013:
307-349). More recent evidence suggests that Descartes may have
continued working on the *Rules* after 1628 (see Descartes ES). *Rules* contains the most detailed description of
Descartes' method anywhere in his corpus. However, he never
completed it, and he never explicitly refers to it anywhere in his
published writings or correspondence. 



Descartes defines "method" in Rule 4 as a set of 



> 
> 
> reliable rules which are easy to apply, and such that if one follows
> them exactly, one will never take what is false to be true or
> fruitlessly expend one's mental efforts, but will gradually and
> constantly increase one's knowledge till one arrives at a true
> understanding of everything within one's capacity. (AT 10:
> 371-372, CSM 1: 16) 
> 
> 
> 



He divides the *Rules* into three principal parts: Rules
1-12 deal with the definition of science, the principal
operations of the method (intuition, deduction, and enumeration), and what Descartes terms "simple propositions", which "occur to us spontaneously" and which are objects of certain and evident cognition or intuition (e.g., "a triangle is bounded by just three lines") (see AT 10: 428, CSM 1: 50; AT 10: 368, CSM 1: 14).
Rules 13-24 deal with what Descartes terms "perfectly
understood problems", or problems in which all of the conditions
relevant to the solution of the problem are known, and which arise principally in
arithmetic and geometry (see AT 10: 429-430, CSM 1: 51); Rules
25-36 deal with "imperfectly understood problems",
or problems in which one or more conditions relevant to the solution of the problem are not
known, but must be found. These problems arise for the most part in
natural philosophy and metaphysics. The *Rules* end prematurely
at Rule 21 (see AT 10: 428-430, CSM 1: 50-51).


## 2. The Method in the *Rules*


### 2.1 Descartes' Definition of "Science"



For Descartes, the sciences are deeply interdependent and
interconnected, and they must be learned by means of one method (AT
10: 360-361, CSM 1: 9-10). He defines
"science" (*scientia*) in Rule 2 as "certain
and evident cognition" (*omnis scientia est cognitio certa et
evidens*, AT 10: 362, CSM 1: 10). Many scholastic Aristotelians
define science in the same way. However, Aristotelians do not believe
that every science satisfies this definition equally; some sciences
(like mathematics) may be more exact and, therefore, more certain than
others (like natural philosophy). Aristotelians consistently make room
for what Descartes terms "probable cognition", especially
in natural philosophy (Rule 2, AT 10: 362, CSM 1: 10). In Rule 2,
Descartes boldly declares that "we reject all [...] merely
probable cognition and resolve to believe only what is perfectly known
and incapable of being doubted" (ibid.).


### 2.2 Intuition



In Rule 3, Descartes introduces the first two operations of the
method: intuition and deduction. He defines "intuition" as
follows:



> 
> 
> By "intuition" I do not mean the fluctuating testimony of
> the senses or the deceptive judgment of the imagination as it botches
> things together, but the conception of a clear and attentive mind,
> which is so easy and distinct that there can be no room for doubt
> about what we are understanding. [...] Thus, everyone can
> mentally intuit that he exists, that he is thinking, that a triangle
> is bounded by just three lines, and a sphere by a single surface, and
> the like. (AT 10: 368, CSM 1: 14)
> 
> 
> 



As Descartes' examples indicate, both contingent propositions
(e.g., that I exist; that I am thinking) and necessary propositions
(e.g., that a triangle is bounded by just three lines; that a sphere
is bounded by a single surface) can be intuited (cf. Alanen and
Yrjonsuuri 1997 and Alanen 1999). Intuition is a type of
intellectual seeing or perception in which the things themselves, not
definitions, are directly present before the mind.
(Descartes chooses the word "intuition" because in Latin
*intueor* means "to look upon, look closely at, gaze
at" and also "to regard, observe, consider, give attention
to".) The evidence of intuition is so direct that
it cannot be doubted. Thus, intuition paradigmatically satisfies
Descartes' definition of science as "certain and evident
cognition". It is the most important operation of the
method.


#### 2.2.1 The Objects of Intuition: The Simple Natures



The principal objects of intuition are "simple natures".
Simple natures are not propositions, but rather notions that are
"so clearly and distinctly [known] that they cannot be divided
by the mind into others which are more distinctly known" (AT 10:
418, CSM 1: 44). The simple natures are, as it were, the atoms of
human knowledge (Hamelin 1921: 86); all other notions and propositions
are *composed* of simple natures. Descartes divides the simple
natures into three classes: intellectual (e.g., knowledge, doubt,
ignorance, volition, etc.), material (e.g., extension, shape, motion,
etc.), and common (e.g., existence, unity, duration, as well as common
notions "whose self-evidence is the basis for all the rational
inferences we make", such as "Things that are the same as
a third thing are the same as each other", etc., AT 10: 419, CSM
1: 45). The intellectual simple natures must be intuited by means of
the intellect alone. The material simple natures must be intuited by
means of the intellect aided by the imagination. The common simple
natures may be intuited either by the intellect alone or the intellect
aided by the imagination (ibid.). The intellectual simple natures
define the essence of mind (one of the objects of Descartes'
metaphysics) and the material simple natures define the essence of
body (the object of Descartes' mathematics and natural
philosophy).



Descartes' theory of simple natures plays an enormously
important role in his method (see Marion 1992). First, the simple natures
"are self-evident and never contain any falsity" (AT 10:
420, CSM 1: 45), and there is nothing in them "beyond what we
intuit or reach in our thinking" (ibid.). Second, 



> 
> 
> it is not possible for us ever to understand anything beyond those
> simple natures and a certain mixture or compounding of one with
> another. (AT 10: 422, CSM 1: 46) 
> 
> 
> 



Third, 



> 
> 
> the whole of human knowledge consists uniquely in our achieving a
> distinct perception of how all these simple natures contribute to the
> composition of other things. (AT 10: 427, CSM 1: 49) 
> 
> 
> 



The theory of simple natures effectively ensures the unrestricted
scope of intuition (and, as I will show below, deduction) vis-a-vis any and all objects
of science, from the simplest to the most complex.


### 2.3 Deduction



Descartes defines deduction as the 



> 
> 
> inference of something as following necessarily from some other
> propositions which are known with certainty [...] provided they
> are inferred from true and known principles through a continuous and
> uninterrupted movement of thought in which each individual proposition
> is clearly intuited. (AT 10: 369, CSM 1: 14-15) 
> 
> 
> 



Descartes provides two useful examples of deduction in Rule 12, where
he writes that "when we deduce that nothing which lacks
extension can have a shape", 



> 
> 
> we intuit that the conjunction of the one with the other is wholly
> necessary [...] on the grounds that there is a necessary
> connection between shape and extension. (AT 10: 424-425, CSM 1:
> 48) 
> 
> 
> 



This "necessary conjunction" is one that I directly "see" whenever I intuit a shape in my
imagination; any shape I imagine will necessarily be extended in
length, width, and breadth. Similarly, if 



> 
> 
> Socrates [...] says that he doubts everything, it necessarily
> follows that he understands at least that he is doubting, and hence
> that he knows that something can be true or false, etc.; for there is
> a necessary connection between these facts and the nature of doubt. (AT
> 10: 421, CSM 1: 46) 
> 
> 
> 



In both of these examples, intuition defines each step of the
deduction, as Descartes requires when he writes that "each
individual proposition" in a deduction must be "clearly
intuited". Deductions, then, are composed of a series or
sequence of intuitions or intuited propositions: 



> 
> 
> Hence we are distinguishing mental intuition from certain deduction on
> the grounds that we are aware of a movement or a sort of sequence in
> the latter but not in the former. (AT 10: 370, CSM 1: 15) 
> 
> 
> 



Just as Descartes rejects Aristotelian definitions as objects of
intuition (Aristotelian definitions like "motion is the actuality of potential being, insofar as it is potential" render motion more, not less, obscure; see AT 10: 426, CSM 1: 49), so too does he reject Aristotelian syllogisms as forms of
deduction or inference (see Gaukroger 1989; Normore 1993; and Cassan
2015). The validity of an Aristotelian syllogism depends exclusively on
its *form*. For example, "All As are Bs; All Bs are Cs; all As
are Cs". Here, no matter what the content, the syllogism remains
valid. For Descartes, by contrast, deduction depends exclusively on
its *content*. What is intuited in deduction are dependency relations between simple natures. I simply
*see* that shape depends on extension, or that doubt depends on
knowledge of the difference between truth and falsity, etc. These
dependencies are immediately revealed in intuition and deduction,
without recourse to syllogistic forms.



Another important difference between Aristotelian and Cartesian
deduction is that Aristotelian deductions *do not yield any new
knowledge*. In the syllogism, "All men are mortal; all Greeks are
men; all Greeks are mortal", the conclusion is already known. We
"learn nothing new from such forms of reasoning" (AT 10:
406, CSM 1: 36). For Descartes, the method "should [...]
extend to the *discovery* of truths in any field
whatever" (AT 10: 374, CSM 1: 17; my emphasis). Descartes'
method is a method of discovery; it does not "explain to others
arguments which are already known".


### 2.4 From Deduction to Enumeration



When deductions are simple, they are wholly reducible to intuition:




> 
> 
> For if we have deduced one fact from another immediately, then
> provided the inference is evident, it already comes under the heading
> of true intuition. (AT 10: 389, CSM 1: 26) 
> 
> 
> 



However, when deductions are "complex and involved" (AT
10: 408, CSM 1: 37) and "we infer a proposition from many
disconnected propositions", then "our intellectual
capacity is often insufficient to enable us to encompass them all in a
single intuition" (AT 10: 389, CSM 1: 26). Fortunately, the
scope of intuition can be expanded by means of an operation Descartes
terms "enumeration". Since some deductions require
"such a long chain of inferences" that it is not
"easy to recall the entire route which led us to" the
conclusion, "a continuous movement of thought is needed to make
good on any weakness of memory" (AT 10: 387, CSM 1: 25). This
involves 



> 
> 
> simultaneously intuiting one relation and passing on to the next,
> until I have learnt to pass from the first to the last so swiftly that
> memory is left with practically no role to play, and I seem to intuit
> the whole thing at once. (AT 10: 287-388, CSM 1: 25) 
> 
> 
> 



Nevertheless, there is a limit to how many relations I can encompass
in a single act of intuition. Were I to continue the series
indefinitely, I would eventually lose track of some of the inferences
in the deductive chain, no matter how many times I traverse the
series. Enumeration is a normative ideal that cannot always be
realized in practice.



Enumeration plays many roles in Descartes' method, and most of
them are not related to the reduction of the role played by memory in
long or complex deductions (see Beck 1952: 111-134; Weber 1964:
48-57; Marion 1975: 103-113; Smith 2010: 67-113).
One can distinguish between five senses of enumeration in the
*Rules*. Enumeration1 is "a verification of
the logical steps already traversed in a deductive process"
(Beck 1952: 143; based on Rule 7, AT 10: 387-388, 14-25,
1-17, CSM 1: 25). Enumeration1 has already been
discussed above.



Enumeration2 is "a preliminary
survey or setting out of the grounds of a demonstration" (Beck
1952: 143; based on Rule 7, AT 10: 388-392, CSM 1: 25-28).
Here, enumeration *precedes* both intuition and deduction.
Enumeration2 determines (a) whatever simpler problems are
contained in a complex problem, and (b) the order in which each of
these problems must be solved, beginning with the simplest problem of
all (for an example, see
 Section 3).



Enumeration3 is "a form of deduction based on the
enumeration of all possible alternatives or analogous instances"
(Beck 1952: 143; based on Rule 7, AT 10: 388-389, 29-30,
1-7, CSM 1: 26 and Rule 8, AT 10: 394-395, CSM 1: 29). For
example, 



> 
> 
> if I wish to show [...] that the rational soul is not corporeal
> [...] it will be sufficient if I group all bodies together into
> several classes so as to demonstrate that the rational soul cannot be
> assigned to any of these. (AT 10: 390, CSM 1: 26-27) 
> 
> 
> 



Here, enumeration is itself a form of deduction: I construct classes
in order to deduce a conclusion. Other examples of
enumeration3 include Descartes' enumeration of his
dubitable opinions in *Meditations* I, which leads to his
discovery in *Meditations* II that he cannot place the
proposition "I am, I exist" in any of these classes (see
 Section 9).



Enumeration4 is "[a]kin to the actual deduction
itself when the implicatory sequence is grounded on a complex and
disjointed set of data" (Beck 1952: 143; based on Rule 7, AT 10:
389, 17-20, CSM 1: 26) (see Beck 1952: 143).
Enumeration4 is a deduction of a conclusion, not from a
series of interconnected inferences, but rather from a variety of
different inferential chains that. For example, Descartes' demonstration that the mind
and body are two really distinct substances in *Meditations* VI
depends on a wide variety of considerations drawn from
*Meditations* I-V (see AT 7: 13, CSM 2: 9; letter to
Mersenne, 24 December 1640, AT 3: 266, CSM 3: 163. For an
interpretation along these lines, see Dubouclez 2013. For a contrary
interpretation, see Gueroult 1984).



Finally, enumeration5 is an operation Descartes also calls
"induction", and consists in an inference from a series of
particular cases satisfying a definite condition to all cases
satisfying the same condition, as when one infers that "the area
of a circle is greater than the area of any other geometrical figure
whose perimeter is the same length as the circle's" from
the fact "this [...] holds for some particular
figures" (AT 10: 390, CSM 1: 27).


## 3. The Method in the *Rules*: An Example



Let's see how intuition, deduction, and enumeration work in
practice. Descartes describes how the method should be applied in Rule
5:



> 
> 
> We shall be following this method exactly if we first reduce
> complicated and obscure propositions step by step to simpler ones, and
> then, starting with the intuition of the simplest ones of all, try to
> ascend through the same steps to a knowledge of all the rest. (AT 10:
> 379, CSM 1: 20)
> 
> 
> 



A clear example of the application of the method can be found in Rule
8, where Descartes discusses how to deduce the shape of the anaclastic
line, i.e., the shape of the lens from which parallel rays of light
are refracted towards a common point, as they are in eyeglasses or
telescopes (see
 Fig. 1).



![Result for anaclastic lens: 3 straight rays heading into and through the flat side of the lens but bend when reaching the curved side to then intersect beyond the curved side and continue](fg-anaclastic.png)

Figure 1: Anaclastic Lens.




The problem of the anaclastic is a complex, imperfectly understood problem. It needs to be
reduced to a ordered series of simpler problems by means of
enumeration2. Once the problem has been reduced to its simplest component parts, the
simplest problem in the series must be solved by means of intuition,
and the more complex problems in the series must be solved by means of
deduction.



Descartes reduces the problem of the anaclastic into a series of five
simpler problems (see Table 1):





---


1. What is the shape of a line (lens) that focuses parallel rays of
light to the same point?
2. What is the relation between angle of incidence and angle of
refraction (i.e., the law of refraction)?
3. How is refraction caused by light passing from one medium to
another?
4. How does a ray of light penetrate a transparent body?
5. What is the nature of the action of light?
6. What is a natural power?



Problem (6) must be solved first by means of intuition, and the
remaining problems must be answered in order:


1. A natural power is...
2. The nature of the action of light is...
3. A ray of light penetrates a transparent body by...
4. Refraction is caused by light passing from one medium to another
when...
5. The relation between the angle of incidence and the angle of
refraction is...
6. The shape of the line (lens) that focuses parallel rays of light
to the same point is...




---



Table 1: Descartes' proposed
deduction of the anaclastic line (Garber 2001: 37)




This example illustrates the procedures involved in Descartes'
method. Intuition and deduction can only performed after
enumeration2 has reduced the problem to an ordered series
of simpler problems. The simplest problem is solved first by means of
intuition, and the more complex problems are solved by means of
deduction. The order of the deduction is read directly off the
enumeration by inversion. Note that identifying some of the
problems in the series (specifically Problems 3-4 in the second
series in
 Table 1)
 require experiment. One must observe how light actually passes
through different types of transparent media in order to determine how
these media affect the angles of incidence and refraction. 


## 4. The Method in *Discourse* II



In Part II of *Discourse on Method* (1637), Descartes offers
the first and only published expose of his method. By
comparison to the method described in the *Rules*, the method described
in *Discourse* II consists of only four rules:



> 
> 
> The first was never to accept anything as true if I did not have
> evident knowledge of its truth: that is, carefully to avoid
> precipitate conclusions and preconceptions, and to include nothing
> more in my judgments than what presented itself to my mind so clearly
> and so distinctly that I had no occasion to doubt it.
> 
> 
> 
> The second, to divide each of the difficulties I examined into as many
> parts as possible and as may be required in order to resolve them
> better.
> 
> 
> 
> The third, to direct my thoughts in an orderly manner, by beginning
> with the simplest and most easily known objects in order to ascend
> little by little, step by step, to knowledge of the most complex, and
> by supposing some order even among objects that have no natural order
> of precedence.
> 
> 
> 
> And the last, throughout to make enumerations so complete, and reviews
> so comprehensive, that I could be sure of leaving nothing out (AT 6:
> 18, CSM 1: 120).
> 
> 
> 



These four rules are best understood as a highly condensed summary of
the method described in the *Rules* (see Gilson 1987: 196-214; Beck 1952: 149; Clarke
1982: 181; Garber 2001: 39; Newman 2019: 85). Descartes'
abridgment of the method in *Discourse* II reflects a shift
vis-a-vis the idea of a "theory" of method. As he
famously put it in a letter to Mersenne, the method consists more in
practice than in theory (letter to Mersenne, 27 February 1637, AT 1:
349, CSMK 3: 53), and to learn the method one should not only reflect
on the rules of the method, but also see how they function in
solutions to particular problems. No matter how detailed a theory of
method may become, there is no way to prepare oneself for every
eventuality that may arise in the course of scientific inquiry, and
there is certainly no way to codify every rule necessary to the
solution of any and all problems. Every problem is different. By the
late 1630s, Descartes decided to reduce the number of rules and focus
on the application of the method rather than on the theory of the
method. In
 Sections 6-9,
 I follow Descartes' advice and examine how he applies the
method in solutions to particular problems in optics, meteorology,
geometry, and metaphysics. First, though, the role played by
experiment in Descartes' method needs to be discussed in more detail.


## 5. Experiment and Supposition in *Discourse* VI and *Principles* III-IV



What role does experiment play in Cartesian science? In *The
World* and *Principles* II, Descartes deduces the
principles of physics (the laws of nature) from the first principle of
metaphysics: God. The laws of nature can be deduced by reason alone
from God's immutability (see AT 11: 36-48, CSM 1:
92-98; AT 8A: 6167, CSM 1: 240-244). Experiment plays
no role in Descartes' deduction of the laws of nature. However,
this does not mean that experiment plays no role in Cartesian science.
On the contrary, in *Discourse* VI, Descartes clearly indicates when experiments become necessary in the course
of scientific inquiry:



> 
> 
> [The] power of nature is so ample and so vast, and these principles
> [sc. the laws of nature] so simple and so general, that I notice
> hardly any particular effect which I do not know at once that it can
> be deduced from the principles in many different ways; and my greatest
> difficulty is usually to discover in which of these ways it depends on
> them. I know no other means to discover this than by seeking further
> observations whose outcomes vary according to which of these ways
> provides the correct explanation (AT 6: 64-65, CSM 1: 144).
> 
> 
> 



There are countless effects in nature that can be deduced from the
laws of nature "in many different ways". How do we find
the *right* way? Experiment. We start with the effects we want
to explain; we isolate and manipulate these effects in order to more
precisely determine the conditions under which they are produced;
and then we make suppositions about what their underlying causes are
based on what we know about the nature of matter and the laws of
nature. Traditional deductive order is reversed; underlying causes too
small to be directly observed are deduced from given effects.
Descartes describes his procedure for deducing causes from effects
toward the end of *Discourse* VI:



> 
> 
> For I take my reasonings to be so closely interconnected that just as
> the last are proved by the first, which are their causes, so the first
> are proved by the last, which are their effects. It must not be
> supposed that I am here committing the fallacy that the logicians call
> "arguing in a circle". For as experience makes most of
> these effects quite certain, the causes from which I deduce them serve
> not so much to prove them as to explain them; indeed, quite to the
> contrary, it is the causes which are proved by the effects. And I have
> called them "suppositions" simply to make it known that I
> think I can deduce them from the primary truths I have expounded
> above. (*Discourse* VI, AT 6: 76, CSM 1: 150)
> 
> 
> 



The suppositions Descartes refers to here are introduced in the course
of experiment; they describe the shapes, sizes, and motions of the
bodies that cause the effects observed in an experiment. Suppositions
are needed because these particles are beyond the reach of
observation. Some scholars have argued that in *Discourse* VI
Descartes introduces a method distinct from the method developed in
the *Rules* and even *Discourse* II. For these scholars, the method in the
*Rules* is *a priori* and proceeds from causes to
effects, while the method in *Discourse* VI is *a
posteriori* and proceeds from effects to causes (see Clarke 1982).
The latter method, they claim, is the so-called
"hypothetico-deductive method" (see Larmore 1980: 6-22 and Clarke 1982:
10). Others have argued that this interpretation of both the
*Rules* and *Discourse* VI suffers from a number of
problems. First, experiment is in no way excluded from the method
developed in the *Rules*. On the contrary, in both the *Rules* and the
*Essays*, experiment neither interrupts nor replaces deduction;
experiment *structures* deduction because it helps one reduce problems to their simplest component parts (see Garber 2001: 85-110). A hint of this
can already be seen in the anaclastic example (see
 Section 3):
 the third problem in the reduction ("How is refraction caused by light passing from one medium to another?") can only be discovered by observing that light behaves
differently in a variety of transparent media.
Experiment structures of the deduction. This is also the case
in Descartes' deduction of the cause of the rainbow (see
 Section 7
 below and Garber 2001: 91-104). Second, in *Discourse* VI,
Descartes explicitly asserts that the suppositions introduced in the
*Essays* can be deduced from first principles or "primary
truths", and there is no room for such demonstrations in the
hypothetico-deductive method, in which hypotheses are confirmed by
experience *alone*.


## 6. The Method in Optics: Deducing the Law of Refraction



Descartes' discovery of the law of refraction is arguably one of
his most celebrated scientific achievements. We have already
encountered the law of refraction in Descartes' discussion of
the anaclastic line in Rule 8 (see
 Section 3).
 There, the law of refraction appears as the solution to the
penultimate problem, "What is the relation (ratio) between the
angle of incidence and the angle of refraction?" We also learned
that the law of refraction depends on two other problems, "What
is a natural power?" and "What is the action of
light?" In *Rules*, Descartes proposes solving the problem of what a natural power is by means of intuition, and he recommends solving the problem of what the action of light consists in by means of deduction or by means of an analogy with other, more familiar natural powers. In Rule 9, analogizes the action of light to the motion of a stick. Similarly,
in *Optics* II, Descartes deduces the law of refraction from
analogies (or comparisons) and suppositions about the reflection and
refraction of light. As in Rule 9, the first comparison analogizes the
action of light to the transmission of motion from one end of a stick
to another, and is meant to illustrate how light travels
instantaneously from one part of space to another:



> 
> 
> I would have you consider the light in bodies we call
> "luminous" to be nothing other than a certain movement, or
> very rapid and lively action, which passes to our eyes through the
> medium of the air and other transparent bodies, just as the movement
> or resistance of the bodies encountered by a blind man passes to his
> hand by means of a stick. (AT 7: 84, CSM 1: 153)
> 
> 
> 



When a blind person employs a stick in order to learn about their
surroundings, they do so via the pressure they receive in their hands
when the stick encounters an object. This resistance or pressure is
instantaneously transmitted from the end of the stick in contact with
the object to the hand. Light, Descartes argues, is transmitted from
the luminous objects to the eye in the same way: it is an
instantaneous pressure exerted on the eye by the luminous object via
the medium (e.g., air). The length of the stick or of the distance
between the sun (or any other luminous object) and our eyes does not
matter, so long as (1) the particles of matter between our hand and
the end of the stick or our eye and the sun are continuous, and (2) the
pressure coming from the end of the stick or the luminous object is
sufficiently strong to affect our hand or eye, so that whatever
happens at one end is instantaneously communicated to the other end
(AT 7: 84, CSM 1: 153).



Descartes' second comparison analogizes (1) the medium in which
light travels to a wine-vat (or barrel) completely filled with
half-pressed grapes and wine, and (2) the action of light in this
medium to the tendency of the wine to move in a straight line towards
holes located at the bottom of the vat:



> 
> 
> The parts of the wine at one place tend to go down in a straight line
> through one hole at the very instant it is opened [...]. In the
> same way, all the parts of the subtle matter [of which light is
> composed] in contact with the side of the sun facing us tend in a
> straight line towards our eyes at the very instant [our eyes] are
> opened [...] (AT 7: 87-88, CSM 1: 154-155).
> 
> 
> 



This comparison illustrates an important distinction between actual
motion from one part of space to another and the mere tendency to
motion. Just as all the parts of the wine in the vat tend to move in a
straight line toward the holes at the bottom of the vat, so too light
is simply a tendency the smallest parts of matter between our eyes and
the sun (or any other luminous object) have to move in a straight line
toward our eye. The space between our eyes and any luminous object is
so crammed that the smallest parts of matter cannot actually travel
from the luminous object to our eye. Instead, their
"action" consists in the tendency they have to move
towards our eyes. This tendency exerts pressure on our eye, and this pressure,
when communicated to the brain via the nerves, produces the sensation
of light in the mind.



The third comparison illustrates how light behaves when its
rectilinear tendency to motion (its tendency to move in a straight
line) is affected by other bodies in reflection and refraction:



> 
> 
> But when [light rays] meet certain other bodies, they are liable to be
> deflected by them, or weakened, in the same way that the movement of a
> ball or stone thrown into the air is deflected by the bodies it
> encounters. For it is very easy to believe that the action or tendency
> to move (which, I have said, should be taken for light) must in this
> respect obey the same laws as motion itself. (AT 7: 88-89,
> CSM 1: 155)
> 
> 
> 



Just as the motion of a ball can be affected by the bodies it
encounters, so too can light be affected by the bodies it encounters.
Depending on how these bodies are themselves physically constituted,
they either reflect or refract light. Soft bodies, such as a linen
sheets, sand, or mud "completely stop the ball and check its
movement", while hard bodies simply "send the ball in
another direction without stopping it" (AT 7: 89, CSM 1: 155).
Since the tendency to motion obeys the same laws as motion itself,
Descartes demonstrates the law of refraction by comparing refracted
light to the motion of a tennis ball before and after it punctures a
linen sheet 



> 
> 
> so thin and finely woven that the ball has enough force to puncture it
> and pass right through, losing only some of its speed (say, a half) in
> doing so. (AT 7: 97, CSM 1: 158; see
>  Fig. 2)
> 
> 
> 



![illustration from Descartes' work showing a man hitting a tennis ball which hits the ground and deflects. Link to extended description and SVG diagram below](fg-tennis.png)

Figure 2: Descartes' tennis-ball
model of refraction (AT 6: 98, CSM 1: 159, D1637: 11 (view 95)). [An
 extended description and SVG diagram of figure 2
 is in the supplement.]




Not every property of the tennis-ball model is relevant to the action
of light, and those that are not relevant can be excluded from
consideration. We can leave aside 



> 
> 
> entirely the question of the power which continues to move [the ball]
> when it is no longer in contact with the racquet, and without
> considering any effect of its weight, size, or shape [...] since
> none of these factors is involved in the action of light. (AT 7:
> 93-94, CSM 1: 157) 
> 
> 
> 



Second, it is necessary to distinguish between the force "which
causes the ball to continue moving" on the one hand, and
"that which determines it to move in one direction rather than
the other" on the other, since "this same force could have
made it move in any other direction" (AT 7: 94, CSM 1: 157).
Conversely, the ball could have been determined to move in the same
direction "even if a different force had moved it"
(ibid.). Third, we can divide the direction of the ball into two
principal components, which determine its direction: a perpendicular
component (line AC) and a parallel component (line AH) (see
 Fig. 2
 above). The ball must be imagined as moving down the perpendicular
line at the same time as it moves across the parallel line (left to
right), and these two components determine its actual
direction along the diagonal (line AB). Descartes terms these components parts of the "determination" of the ball because they specify its direction. Determinations are directed physical magnitudes. As we will see below, they specify the direction of the ball, and they can be independently affected in physical interactions.



Descartes proceeds to deduce the law of refraction. The ball is struck
by the racquet at A and moves along AB until it strikes the sheet at
B. Where will the ball land after it strikes the sheet? Descartes
stipulates that the sheet reduces the speed of the ball by half.
Consequently, it will take the ball twice as long to reach the
circumference of the circle after impact than it did for the ball to
reach the surface at B. We also know that the determination of the
ball in direction AB is composed of two parts, a perpendicular
component determination (AC) and a parallel component determination (AH). Descartes reasons that




> 
> 
> only the one [component determination] which was making the ball tend in a downward
> direction [AC] can be changed in any way through its colliding with
> the sheet, while the one which was making the ball tend to the right
> [AH] must always remain the same as it was, because the sheet offers
> no opposition at all to the determination in this direction. (AT 7:
> 97, CSM 1: 159) 
> 
> 
> 



To understand Descartes' reasoning here, the parallel component
determination AH must be regarded as simply continuing along its initial path
because it does not come into contact with the surface of the sheet.
To where must AH be extended? Since the ball has lost half of its
initial speed and consequently will take twice as long to reach the
circumference of the circle after impact, we double the length of AH
by extending it to F. The ball must, therefore, land somewhere on the
line dropped from F, but since it cannot land above the surface, it
must land somewhere below CBE. It lands precisely where the line
dropped from F intersects the circle at I (ibid.). Once we have I, we
extend AB to I. Descartes observes that the degree of refraction
"varies exactly in proportion to the varying degrees of
penetrability of the respective bodies" (AT 7: 101, CSM 1: 161).
He insists, however, that the quantities that should be compared to
one another in this proportion are not the angles ABH and IBE
themselves (the angles of incidence and refraction, respectively),
"for the ratio or proportion between these angles varies with
all the different inclinations of the rays" (ibid.). In other
words, the angles of incidence and refraction do not vary according to
any determinable proportion. Instead of comparing the angles to one
another, Descartes compares the lines AH and HF (the sines of the angles of incidence and refraction, respectively), and sees
that the proportion between these lines is that of 1/2, a ratio that
all refractions between these two media, whatever the angles of
incidence and refraction, must obey. Since the lines AH and HF are the
sines of the angles, Descartes' law of refraction is oftentimes
referred to as the "sine law".

 

\[\sin i = n \sin r \]


The sine of the angle of incidence *i* is equal to the sine of
the angle of refraction *r* multiplied by a constant *n*
defined by the nature of the refractive medium (in the example
discussed above, the constant defined by the sheet is 1/2 , so AH =
1/2 HF).



Many commentators have raised questions about Descartes'
deduction of the sine law (see, e.g., Schuster 2013: 178-184).
For example, what physical meaning do the parallel and perpendicular
component determinations (lines AH and AC) have? Is it really the case that the
speed of the ball is reduced only at the surface of impact, and not
after (see Schuster 2013: 180-181)? In water, it would seem that the speed of the ball is reduced as it penetrates further into the medium. These and other questions
cannot be examined in detail here. From a methodological point of
view, Descartes insists that the law of refraction can be deduced from
the comparisons and suppositions he employs in *Optics* II (see letter to
Mersenne, 27 May 1638, AT 2: 142-143, CSM 1: 103), and as we have seen, in both Rule 8 and *Discourse* IV he claims that he can demonstrate these suppositions from the principles of physics.


## 7. The Method in Meteorology: Deducing the Cause of the Rainbow



Descartes' deduction of the cause of the rainbow in
*Meteorology* VIII has long been regarded as one of his
clearest applications of the method (see Garber 2001: 85-110).
Descartes himself seems to have believed so too (see AT 1: 559, CSM 1:
85). In *Meteorology* VIII, Descartes explicitly points out
that he could not have chosen 



> 
> 
> a more appropriate subject for demonstrating how, with the method I am
> using, we can arrive at knowledge not possessed at all by those whose
> writings are available to us. (AT 6: 325, MOGM: 332)
> 
> 
> 



Descartes begins his inquiry into the cause of the rainbow by
enumerating2 all of the conditions relevant to the solution of the problem, beginning with when and where rainbows appear in nature.
Rainbows appear 



> 
> 
> not only in the sky, but also in the air near us, whenever there are
> many drops of water in the air illuminated by the sun, as experience
> shows us in certain fountains. (AT 6: 325, MOGM: 332) 
> 
> 
> 



This observation yields a first conclusion: 



> 
> 
> [Thus] it was easy for me to judge that [the rainbow] came merely from
> the way that the rays of light act against those drops, and from there
> toward our eyes. (AT 6: 325, CSM 1: 332) 
> 
> 
> 



Drawing on his earlier description of the shape of water droplets in
*Meteorology* V (AT 6: 279-280, MOGM: 298-299),
Descartes reasons that 



> 
> 
> knowing that these drops are round, as has been proven above, and
> seeing that their being larger or smaller does not change the
> appearance of the arc, I then took it into my head to make a very
> large one, the better to examine it. (AT 6: 280, MOGM: 332) 
> 
> 
> 



He designs a *model* that will enable him to acquire more
observations about of the behavior of light when it acts on water.
Since water is perfectly round, and since the size of the water does
not change the appearance of the arc, he fills "a perfectly
round and transparent large flask with water" and examines the
behavior of light when it acts on the water in the flask. Already at
this early stage, delicate considerations of relevance and irrelevance
are clearly on display, and these considerations allow Descartes to
narrow down and more clearly define the problem.
Once he filled the large flask with water, he 



> 
> 
> discovered that, for example, when the sun came from the section of
> the sky marked AFZ, and my eye was at point E, then when I put this
> ball in the location BCD, its part D appeared to me completely red and
> incomparably more brilliant than the rest [...]. (AT 6:
> 325-326, MOGM: 332; see
>  Fig. 3).
> 
> 
> 



![link to extended description below](fg-flask.png)

Figure 3: Descartes' flask model
of sunlight acting on water droplets (MOGM: 333). [An
 extended description and SVG diagram of figure 3
 is in the supplement.]




As Descartes surely knew from experience, red is the last color of the
primary rainbow (located in the uppermost section of the bow) and the
first color of the secondary rainbow (located in the lowermost section
of the bow). As he also must have known from experience, the red in
the primary rainbow is much brighter than the red in the secondary
rainbow. Consequently, Descartes' observation that D appeared
completely red and more brilliant than all other parts of the flask
must have immediately struck him as significant and promising. He
therefore proceeded to explore the relation between the rays of the
sun, the position of his eyes, and the brightness of the red at D by
varying the conditions, observing what changes and what remains the
same in order to more precisely determine the relevant factors.
The brightness of the red at D is not affected by placing the flask to
the right or to the left of the observer, nor by the observer turning
round the flask, so long as the angle DEM remains the same. When
Descartes measures it, the angle DEM is 42o. Therefore, it is the
42o angle the eye makes with D and M at DEM alone that plays a
role in the appearance of the brighter red at D. Having identified the
measure of angle DEM, Descartes then varies the angle in order to
determine what other changes, if any, occur. This enables him to
correlate the decrease in the angle to the appearance of other colors
in the flask: 



> 
> 
> And if I made the angle slightly smaller, the color did not appear all
> at once, but rather it first divided into two less brilliant parts, in
> which one saw yellow, blue, and other colors 
> 
> 
> 



of the primary rainbow (AT 6: 326-327, MOGM: 333).



Descartes then turns his attention toward point K in the flask, and
observes that 



> 
> 
> if I made the angle KEM around 52o, this part K would appear red
> too, but not as brilliant as at D; and that if I made it slightly
> larger, other weaker colors would appear. But I found that if I made
> it ever so slightly smaller, or very much larger, no colors would
> appear. (ibid.) 
> 
> 
> 



Once more, Descartes identifies the angle at which the less brilliant
red appears, this time at K, closer to the top of the flask, and
observes that, by slightly enlarging the angle, other, weaker colors
appear, as they do in the secondary rainbow. He concludes, based on
these observations, that if the air were filled with drops of water,
these drops would produce the same colors, relative to the same
angles, effectively producing all the colors of the primary and
secondary rainbows.



So far, considerable progress has been made. The angles at which the
colors of the primary and secondary rainbows appear have been
determined. We have acquired more precise information about when and
where rainbows appear. However, we do not yet have an explanation. The
underlying cause of the rainbow remains unknown. Thus, Descartes
decides to examine "in more detail what caused the part D of the
ball BCD to appear red", and finds that 



> 
> 
> it was the rays of the sun which, coming from A toward B, were curved
> [refracted] as the entered the water at point B, and went toward C,
> whence they were reflected toward D; and there, being curved
> [refracted] again as they left the water, they tended toward E. 
> 
> 
> 



How did Descartes arrive at this particular finding? By
*intervening directly in the model* in order to exclude factors
irrelevant to the production of the effect (the bright red at D) and
cleanly isolate the cause that alone produces it. Whenever he
"put an opaque or dark body in some place on the lines AB, BC,
CD, or DE, this red color would disappear", but whenever he
"covered the whole ball except for the points B and D, and put
dark bodies everywhere else", then the red color would appear at
D. Similarly, in the case of K, he discovered that the ray that
produces the red color there comes from F toward G, where it is
refracted toward H, and thence reflected toward I, and at I once more
reflected, this time toward K, where it is refracted toward E. He
concludes: 



> 
> 
> Therefore the primary rainbow is caused by the rays which reach the
> eye after two refractions and one reflection, and the secondary by
> other rays which reach it only after two refractions and two
> reflections; which is what prevents the second from appearing as
> clearly as the first. (AT 6: 328-329, MOGM: 334) 
> 
> 
> 



(As we will see below, another experiment Descartes conducts reveals
that this conclusion is false, and that only one refraction is needed
to produce the colors of the rainbow. It is interesting that Descartes
would choose to include a result he will later overturn.)



Descartes next examines what he describes as the "principal
difficulty". Suppose a ray strikes the flask somewhere between K
and B, undergoes two refractions and one or two reflections, and upon
leaving the flask tends toward the eye at E. Why this ray produces no
color, and "only those of which I have spoken [...] cause
certain colors to appear", is not clear (AT 6: 329, MOGM: 334).
The difficulty here is twofold. First, why is it that only the rays
Descartes has identified produce colors? Second, why do these rays
produce "certain colors", i.e.., these colors in this
particular order (see Buchwald 2008: 10)? To resolve this difficulty,
Descartes 



> 
> 
> looked to see if there were some other subject where they [the
> colors] appeared in the same way, so that by comparing them with each
> other I could better judge their cause. (AT 6: 329, MOGM: 335) 
> 
> 
> 



The principal function of the comparison is to determine whether the factors
that produce the colors of the rainbow in water can be found in other
media. Descartes decides to examine the production of these colors in
a prism (see
 Fig. 4).
 Prisms are differently shaped than water, produce the colors of the
rainbow without any reflections, and with only one refraction. These
conditions are rather different than the conditions in which the
colors of the rainbow are produced in a flask. Clearly, then, the true
cause of the rainbow has not yet been fully determined.



![Image of Descartes prism. link to extended description below](fg-sun.png)

Figure 4: Descartes' prism model
(AT 6: 330, MOGM: 335, D1637: 255). [An
 extended description and SVG diagram of figure 4
 is in the supplement.]




In the prism model, the rays emanating from the sun at ABC "cross MN at
right angles, or nearly so, so that they do not undergo any noticeable
refraction there", but "suffer a fairly great refraction
in coming out through NP" (AT 6: 329-330, MOGM: 335).
Furthermore, it is only when the two sides of the bottom of the prism
NP are covered by a dark body of some sort, so that the rays could
only exit through the narrow opening at DE, that the rays paint all
the colors of the rainbow on the cloth or white paper FGH, always
producing red at F, and blue or violet at H (ibid.). By comparing
differences between the flask and the prism, Descartes learns
"that the surfaces of the drops of water need not be curved in
order to produce these colors, for those of this crystal are
completely flat". He also learns that "the angle under
which they appear need not be any particular size, for it can be
changed here without their changing" (ibid.). Here, Descartes is
referring to the angle of refraction (e.g., HEP), which can vary
between the flask and the prism and yet produce the same effect, and
which can also be the same for rays ABC in the prism at DE and yet
produce different colors at FGH. He further learns that 



> 
> 
> neither is reflection necessary, for there is none of it here; nor
> finally do we need a plurality of refractions, for there is only one
> of them here. (ibid.) 
> 
> 
> 



He also "had no doubt that light was necessary, for without it
we would see nothing" (AT 6: 331, MOGM: 335). Finally, he 



> 
> 
> observed [...] that shadow, or the limitation of this light, was
> necessary; for if we remove the dark body on NP, the colors FGH cease
> to appear, and if we make the opening DE large enough, the red,
> orange, and yellow at F extend no further because of that than do the
> green, blue, and violet at H--instead, all the extra space
> between the two at G remains white. (AT 6: 331, MOGM: 336) 
> 
> 
> 



When the dark body covering two parts of the base of the prism is
completely removed, no colors appear at all at FGH, and if it is
opened too widely, all of the colors retreat to F and H, and no colors
appear in between (see Buchwald 2008: 14).



Descartes has so far compared the production of the rainbow in two
distinct models: the flask and the prism. In both cases, he enumerates
variations and invariances in the production of one and the same
effect, excludes irrelevant causes, and pinpoints only those that are
necessary. Interestingly, the second experiment in particular also
raises *new problems*, problems Descartes could not have been
familiar with prior to the experiment, but which do enable him to more
concretely define the series of problems he needs to solve in order to
determine the cause of the rainbow (see Garber 2001: 101-104 and
Buchwald 2008). For example, the colors produced at F and H (see
 Fig. 6
 below) are different, "even though the refraction, shadow, and
light concur there in the same way" (AT 6: 331, MOGM: 336). What
causes these colors to differ?



To solve this problem, Descartes draws
on his previous research in *Optics* and reflects on the nature
of the particles whose motions at the micro-mechanical level, beyond
what can be observed by the senses, produce visible light. Descartes' method
requires that every phenomenon in nature be reducible to the material
simple natures of extension, shape, and motion (see
 Section 2.2.1
 above). What remains to be determined in this case is "what
sort of mixture of simple natures is necessary for producing all the
effects" of the rainbow (AT 10: 427, CSM 1: 49), i.e., how the
extension, shape, and motion of the particles of light produce the
precise order of the colors of the rainbow. The transition from the
prism to the micro-mechanical level is naturally prompted by the fact
that neither the flask nor the prism can be of any assistance in
finding the cause of the order of the colors of the rainbow. The prism
only provides conditions in which "the refraction, shadow, and
light concur in the same way" and yet produce different colors
in different places on FGH. The cause of the color order cannot be
disclosed by the mere examination of the models.



In *Optics*, Descartes described the nature of light as 



> 
> 
> the action or movement of a certain very fine material whose particles
> must be pictured as small balls rolling in the pores of earthly bodies
> (AT 6: 331, MOGM: 336) 
> 
> 
> 



The manner in which these balls tend to rotate depends on the causes
that determine them to do so. When they are refracted by a common
surface, "all the refractions which occur on the same side [of
the balls] cause them to turn in the same direction" (ibid.),
but they do not necessarily have the same tendency to rotational
speed. Particles of light can acquire different tendencies to
rotational speed after refraction, depending on the bodies that
surround them, 



> 
> 
> so that those which have a much stronger tendency to rotate cause the
> color red, and those which have only a slightly stronger tendency
> cause yellow, 
> 
> 
> 



while 



> 
> 
> the nature of those that are visible at H consists only in the fact
> that these small particles do not rotate as quickly as they usually do
> [...] so that green appears when they turn just a little more
> slowly, and blue where they turn very much more slowly. (ibid.) 
> 
> 
> 



The balls that compose the ray EH have a weaker tendency to rotate,
while those that compose the ray DF have a stronger one. Differences
in color are therefore produced by differential tendencies to
rotational speed after refraction. This is a characteristic example of
how mechanical explanation in Cartesian natural philosophy operates.
The description of the behavior of particles at the micro-mechanical
level explain the observable effects of the relevant phenomenon.



The rays coming toward the eye at E are clustered at definite angles
in the flask, and these angles determine which rays reach our eyes and
which rays do not (see
 Fig. 5).



![Image from Descartes' book; link to extended description and SVG diagram below](fg-rainbow.png)

Figure 5 (AT 6: 328, D1637: 251). [An
 extended description and SVG diagram of figure 5
 is in the supplement.]




At DEM, which has an angle of 42o, the red of the primary rainbow
appears, and below it, at slightly smaller angles, appear the
remaining colors of the primary rainbow (orange, yellow, green, blue,
violet). At KEM, which has an angle of about 52o, the fainter red
of the secondary rainbow appears, and above it, at slightly larger
angles, appear the remaining colors of the secondary rainbow (orange,
yellow, green, blue, violet). The neighborhood of the two principal
angles DEM and KEM alone receive a sufficient number of rays to
produce all the colors of the primary and secondary rainbows. Beyond
them, there lies only "shadow", i.e., light rays that, due
to their small number, produce no color. The conditions under which
colors are produced in the prism do indeed faithfully reproduce those
in which the colors of the rainbow are naturally produced, and
together the flask, the prism, and Descartes' physics of light
deduce all of the effects of the rainbow.



The structure of the deduction is exhibited in
 Figure 6.



![a diagram: link to extended description below](fg-rainbow-deduction.svg)

Figure 6: Descartes' deduction of
the rainbow (Garber 2001: 100). [An
 extended description of figure 6
 is in the supplement.]



## 8. The Method in Mathematics



In a letter to Mersenne written toward the end of December 1637,
Descartes intimates that 



> 
> 
> [in] the *Optics* and the *Meteorology* I merely tried
> to show that my method is better than the usual one; in my
> *Geometry*, however, I claim to have demonstrated this. (AT 1:
> 478, CSMK 3: 77-78) 
> 
> 
> 



Geometrical problems are perfectly understood problems; all the
conditions needed to solve the problem are provided in the statement
of the problem (see
 Section 1). To solve any problem in geometry, one must find a
line(s) that bears a definite relation to given lines. 



> 
> 
> All the problems of geometry can easily be reduced to such terms that
> thereafter we need to know only the length of certain straight lines
> in order to construct them. (AT 6: 369, MOGM: 177) 
> 
> 
> 



These lines can only be found by means of the addition, subtraction,
multiplication, division, and root extraction of given lines. While it
is clear how these operations can be performed on numbers, it is less
clear how they can be performed on lines. What, for example, does it
mean to multiply one line by another?



A very elementary example of how multiplication may be performed on
lines can be seen in the problem of squaring a line. Let line *a*
be the given line, and let it be required to multiply *a* by itself
\((x=a^2).\) To find the value of *x*, I simply construct the
square \(a^2\) below (see
 Fig. 7):





![a line ](fg-lsc-a-line.svg)

\(a\)




![a square ](fg-lsc-b-square.svg)

\(a^2\)




![a cube ](fg-lsc-c-cube.svg)

\(a^3\)





Figure 7: Line, square, and cube. The
problem of dimensionality.




This example clearly illustrates how multiplication may be performed
on lines, but its simplicity conceals a problem. Descartes'
predecessors regarded geometrical constructions of arithmetical
operations in an extremely limited way: due to the fact that in
geometry there are only three spatial dimensions, multiplication
beyond the cube proved difficult. A number can be represented by a
line, the square of a number by a surface (a square), and the cube of
a number by a solid (a cube), but beyond the solid, there are no more
dimensions in which to represent the multiplication of \(n > 3\)
lines (see Mancosu 2008: 112) (see
 Fig. 7).
 The "problem of dimensionality", as it has since come to
be known, constituted a serious obstacle to the use of algebra in
geometry (ibid.). Suppose the problem is to raise a line to the fourth
power \((x=a^4).\) For Descartes' predecessors, this made
absolutely no geometrical sense.



Descartes solved the problem of dimensionality by showing how
arithmetical operations performed on lines never transcend the line.
For example, if line AB is the unit (see
 Fig. 8),
 and I want to multiply line BD by BC, "I have only to join the
points A and C, then to draw DE parallel CA, and BE is the product of
this multiplication" (AT 6: 370, MOGM: 177-178).
Descartes' procedure is modeled on similar triangles (two or
more triangles whose sides may have different lengths but whose angles are equal). The sides of all similar
triangles are proportional to one another (e.g., triangle ACB is
similar to triangle DEB, such that BC is proportional to BE and BA is
proportional to BD, etc.) (see Euclid's
 *Elements* VI.4-5
 [1908: [2] 200-204]). By exploiting the theory of proportions,
Descartes could easily show that BA:BD=BC:BE, or \(1:a=b:c\) (e.g.,
\(1:2=2:4,\) so that \(2*2=4,\) etc.), in which case
\(a*b=c\) or \(\textrm{BD}*\textrm{BC}=\textrm{BE}.\) The
multiplication of two or more lines never produces a square or a
solid, but only another line segment that bears a definite
(proportional) relation to the other line segments. All magnitudes can
easily be compared to one another as lines related to one another by
some measure or proportion, effectively opening the door to the
unrestricted use of algebra in geometry.



![Image from Descartes work. Two lines meet at point B to form an acute angle; points D and A with D further from B are on the lower line and points E and C with point E further from B are on the upper line; a line segment connects D and E and also A and C. Link to SVG diagram and extended description below](fg-two-lines.png)

Figure 8 (AT 6: 370, MOGM: 178, D1637:
298). [An
 extended description and SVG diagram of figure 8
 is in the supplement.]




Having explained how multiplication and other arithmetical operations
can be employed in geometry (AT 6: 369-370, MOGM:
177-178), Descartes proceeds to describe how the method should
be applied to problems in geometry:



> 
> 
> Thus, if we wish to solve some problem, we should first of all
> consider it solved, and give names to all the lines--the unknown
> ones as well as the others--which seem necessary in order to
> construct it. Then, without considering any difference between the
> known and the unknown lines, we should go through the problem in the
> order which most naturally shows the mutual dependency between these
> lines, until we have found a means of expressing a single quantity in
> two ways. This will be called an equation, for the terms of one of the
> two ways [of expressing the quantity] are equal to those of the other.
> (AT 6: 372, MOGM: 179)
> 
> 
> 



This is the method of analysis, which will also find some application
in metaphysics (see
 Section 9).
 To apply the method to problems in geometry, one must first
"consider [the problem] solved", using letters to name
both known and unknown lines. One must then produce as many equations
as there are unknown lines, and each equation must express the unknown
line in terms of the known lines. (Equations define unknown magnitudes
in terms of known magnitudes. For example, the equation \(x^2=ax+b^2\)
defines the unknown magnitude "*x*" in relation to
the known magnitudes "*a*" and
"*b*", thereby expressing one quantity in two ways.)
Finally, one must employ these equations in order to geometrically
construct the required line(s).



Descartes provides an easy example in *Geometry* I. The line
segments *a* and *b* are given, and I must construct a line
*x* such that \(x^2 = ax+b^2.\) The construction proceeds as
follows (see
 Fig. 9).




![Image from Descartes. A circle (dotted) with a center of N and bisected by the line segment OP (with O at approximately 160 degrees and P at -20 degrees). Point L is on the circle at -90 degrees from N. The line segment OP extends beyond the circle to intersect at point M with a horizontal line extending from L. Link to SVG diagram and description below](fg-circle.png)




Figure 9 (AT 6: 375, MOGM: 181, D1637:
302). [An
 extended description and SVG diagram of figure 9
 is in the supplement.]




First, I draw a right-angled triangle NLM, such that \(\textrm{LN} =
1/2 a\), \(\textrm{LM} = b\) and the angle \(\textrm{NLM} =
90o.\)



Second, I draw a circle with center N and radius \(1/2a\).



Third, I prolong NM so that it intersects the circle in O.



MO = *x*, the required line segment.



Proof: By *Elements* III.36,
\(\textrm{MO}*\textrm{MP}=\textrm{LM}^2.\) Therefore,
\(x(x-a)=b^2\) or \(x^2=ax+b^2\) (see Bos 2001: 305).



This procedure is relatively elementary (readers not familiar with the
relevant Euclidean constructions are encouraged to consult
Euclid's
 *Elements* III.36
 [1908: [2] 73-75]). The method employed is clear. The problem
is algebraically expressed by means of letters for known and unknown
magnitudes, and an equation is produced in which the unknown magnitude
is expressed exclusively in terms of known magnitudes. The unknown
magnitude is then constructed by the addition of a line that satisfies
the equation. The construction is such that the solution to the
problem can be intuited or directly *seen* in spatial
intuition by the intellect aided by the imagination (or on paper,
which embodies the operations of the intellect on line segments in the
imagination). Geometrical construction is, therefore, the foundation
of intuition in Cartesian geometry, and it constitutes the final step
in the solution to any problem.



Descartes employed his method in order to solve problems that had
never been solved in the history of mathematics. One such problem is
the "Pappus problem", a locus problem, or problem in which
one must find the locus (location) of all points satisfying a definite
condition (equation), stated by the fourth-century Greek mathematician
Pappus of Alexandria (c. 300-350): 



> 
> 
> [If] we have three, or four, or a greater number of straight lines
> given in position, we must first of all have a point from which we can
> draw as many other straight lines, one on each of the given lines,
> which form given angles with them. (AT 6: 379, MOGM: 184) 
> 
> 
> 



such that a definite ratio between these lines obtains. In the case of
locus problems involving more than six lines (in which three lines on
one side of the equation must be shown to have a proportional relation
to four lines on the other side), Pappus believed that the problem of
dimensionality prohibited solutions to these problems, "since
there is no figure of more than three dimensions", so that
"a figure contained by these lines is not understandable in any
way" (ibid.). For Descartes, by contrast, geometrical sense can
be made of the multiplication of any number of lines. Descartes
provides a completely general solution to the Pappus problem: no
matter how many lines, he demonstrates how it is possible to find an
equation and produce a construction satisfying the required conditions
(see Bos 2001: 313-334). More broadly, he provides a complete
enumeration of the types of problem one encounters in geometry
(defined by degree of complexity); enumerates the geometrical
constructions required to solve problems in each class; and defines
the class of geometrically acceptable constructions by whether or not
they can be algebraically expressed.


## 9. The Method in Metaphysics



Descartes employs the method of analysis in *Meditations*
(Second Replies, AT 7: 155-156, CSM 2: 110-111). In
metaphysics, the method of analysis "shows how the thing in
question was discovered" (ibid.). Descartes opposes analysis to
synthesis, in which first principles are not discovered, but rather
given in the form of definitions, postulates, axioms, theorems, and
problems (ibid.), as in a Euclidean demonstrations. Synthesis
"cannot so conveniently be applied to [...] metaphysical
subjects", Descartes writes. Why? 



> 
> 
> The difference is that the primary notions which are presupposed for
> the demonstration of geometrical truths are readily accepted by
> anyone, since they accord with the use of our senses. [...] In
> metaphysics by contrast there is nothing which causes so much effort
> as making our perception of the primary notions clear and distinct.
> (AT 7: 156-157, CSM 1: 111) 
> 
> 
> 



In metaphysics, the first principles are not provided in advance,
because the mind must be habituated or learn how to perceive them
clearly and distinctly, and habituation requires preparation (the
rejection of preconceived opinions and the perfected employment of the
cognitive faculties). Furthermore, the principles of metaphysics must
be indubitable, and since their indubitability cannot be assumed, it
must be shown. 



Descartes' analytical procedure in *Meditations* I
consists in enumerating3 his opinions and subjecting them
to doubt, so that any proposition that survives these doubts can be
securely accepted as true. Descartes also describes this as the
"method of universal doubt" (AT 7: 203, CSM 2: 207). 



> 
> 
> [For] the purpose of rejecting all my opinions, it will be enough if I
> find in each of them at least some reason for doubt. And to do this I
> will not need to run through them all individually, which would be an
> endless task. [...] I will go straight for the principles. (AT 7:
> 18, CSM 2: 17) 
> 
> 
> 



Instead of running through all of his opinions individually, he
decides to place them in definite classes and examine one or two
members of each particular class, in order to see whether he has any
reason to doubt them. He defines the class of his opinions as those
"I have acquired either from the senses or through the
senses" (AT 7: 18, CSM 1: 12) and proceeds to further divide the
class into (a) opinions about things "which are very small or in
the distance", about which he frequently errs; (b) opinions
about his body and things that are in his immediate environment, which
may be little more than a dream; (c) opinions about things, which even
if they are imaginary, are at least fashioned out of things that are
real, a 



> 
> 
> class [which] appears to include corporeal nature in general, and its
> extension; the shape of extended things; the quantity, or size and
> number of these things; the place in which they may exist; the time
> through which they may endure, and so on. 
> 
> 
> 



He then doubts the existence of even these things, since there may be
a God who 



> 
> 
> brought it about that there is no earth, no sky, no extended thing, no
> shape, no size, no place, while at the same time ensuring that all
> these things appear to me to exist just as they do now. (AT 7:
> 18-21, CSM 2: 12-14) 
> 
> 
> 



Descartes completes the enumeration of his opinions in
*Meditations* I by concluding that 



> 
> 
> I have no answer to these arguments, but am finally compelled to admit
> that there is not one of my former beliefs about which a doubt may not
> properly be raised. [...] So in future I must withhold my assent
> from these former beliefs just as carefully as I would from obvious
> falsehoods, if I want to discover any certainty. (AT 7: 21-22,
> CSM 2: 14-15) 
> 
> 
> 



The doubts entertained in *Meditations* I are entirely structured by
enumeration3 (see Descartes' remarks on enumeration
in Rule 7, AT 10: 391, CSM 1: 27 and
 Section 2.4
 above).



The famous intuition of the proposition, "I am, I exist"
in *Meditations* II is discovered by means of
enumeration3: the proposition "I am, I exist",
cannot be placed into any of the classes of dubitable opinions
enumerated in *Meditations* I because not even the most
malicious demon "can bring it about that I am nothing so long as
I think that I am something" (AT 7: 25, CSM 2: 17). Here,
intuition comes after enumeration3 has prepared the
way.



These examples show that enumeration both orders and enables Descartes
to solve a variety of problems in *Meditations* (see
 Section 2.4
 above and Dubouclez 2013: 307-331). Intuition and deduction are
mobilized only after enumeration has prepared the way. Descartes
effectively deals with a series of imperfectly understood problems in
*Meditations*, and he solves these problems by means of three
operations: enumeration (principally enumeration2-4),
intuition, and deduction.



Not everyone agrees that the method employed in *Meditations*
is the method described in the *Discourse* and the
*Rules*. Some scholars have very plausibly argued that the
"method of doubt" in *Meditations* constitutes a
distinct method. In *Meditations*, Descartes actively resolves
"to doubt all previous beliefs by searching for grounds of
doubt" (Curley 1978: 43-44; cf. Broughton 2002: 2-7). Descartes does
not resolve to doubt all of his former opinions in the *Rules*.
Furthermore, in the case of the anaclastic, the method of the
*Rules* requires reducing complex problems to a series of
simpler problems; solving the simplest problem by means of intuition;
and solving the more complex problems by means of deduction (see
 Section 3).
 It is difficult to discern any such procedure in *Meditations*
(Garber 1992: 49-50 and 2001: 44-47; Newman 2019). Other
scholars have argued that Descartes' method in the
*Rules* does play an important role in *Meditations*.
Descartes' metaphysical principles are discovered by combining
simple natures, such as the combination of thought and existence in
the performance of the *cogito* in *Discourse* IV and
*Meditations* II (see Marion 1992 and the examples of intuition discussed in
 Section 2.2
 above). A recent line of interpretation maintains more broadly that
Descartes' method can be applied in different ways. Different
types of problems must be solved differently (Dika and Kambouchner
forthcoming). The method of doubt is not a distinct method, but rather
an application of the same method to a different problem.