FIVE
Order, Natures, and Series The Topology of Imagination

Rule 4 left us in the following situation. To actualize the fundamental unity of all science we need not just the two ways of knowing, intuitus and deductio, but also a methodical way of approaching and acquiring knowledge. Descartes vaunts himself as the first not so much to realize that there is such a method (for he finds traces of it in predecessors) as to explicate and publicize it. Rule 4 told how he came to this discovery and explained (l) that this method of mathesis is a discipline for which the hitherto most certain of sciences, the mathematical, are like vestments, (2) that it is the source of what is perspicuous and easy in ordinary mathematics, (3) that the mathematicians of his day who cultivate algebra are attempting to resuscitate the mathetical art of analysis that the ancients knew, and (4) that this mathesis is the science of everything having to do with order and measure. The art of analysis of the ancients took both what was known and what was sought in a problem as given and used the codetermination of the parts to arrive at an exact determination of the unknown.^[1] In general, the values that are possible for any one of a set of factors are at least partially determined and delimited by the structure of relationships that hold between them all.

If we recall that the Regulae was projected to consist of three parts, the latter two being devoted to perfect and imperfect problems, respectively, we begin to see more clearly how in principle the account of the method was to be laid out. After Part I set down the basics of the doctrine, Part II would show how the codetermination of factors could be used to discover unique solutions for unknowns, and Part III would deal with situations

[1] See chap. 4, Sec. G, above.

where the givens are insufficient to determine a solution.^[2] We see that the lines and rectangles that Descartes introduced in Rules 15 through 21 are meant to create a kind of calculus, an algebra using figures rather than numbers, to instantiate the dimensions of a problem. The theory of proportions that Descartes had been working on at least since 1619 received its ultimate methodological expression in this geometric calculus.

What harm is there, then, in saying that this science, this universal mathesis, is mathematics, that the Regulae creates a new, more general type of mathematics to deal with problems of all kinds at a high level of abstraction, and thus that the Regulae culminates in the doctrine of the universal applicability of mathematics? After all, the Regulae in essence teaches how mathematical problems can be presented through a system of equations that can be turned into geometry; these equations, whether in one or several unknowns, and whatever their degree, are solvable precisely insofar as the various terms are codetermined by their relationships to one another.

One difficulty with this conception is that it might simply be an anachronism deriving from the long tradition of historiography of science that credits Descartes with inventing modern analytic geometry. What that tradition tends to minimize is that Descartes's approach was motivated by geometrical considerations more than by what we consider algebra.^[3] But this orientation to geometry, on my interpretation, is due not so much to an essentially backward-looking (and, from a twentieth-century viewpoint, atavistic) notion of geometry's preeminence over arithmetic (and thus also over algebra) as it is to the method of imaginative cognition that Descartes had been cultivating at least since 1618, a method that gives primacy to the formation and use of images. What the Regulae is attempting to pro-

[2] Rule 8 presents the distinction between the parts somewhat differently, however: it promises that the second part will deal with composite natures that are deduced from the simplest natures, and the third with composite natures "that presuppose other things also, which we experience to be composite on the part of the thing" (AT X 399).

[3] It is unclear at what stage in his career Descartes began using—and apparently invented—a modem power notation for representing polynomials. Milhaud dates Descartes's first use of a modern notation to around 1620; see Milhaud, Descartes savant, 86. P.J. Fededco places Descartes's invention of modern notation close to 1630; see Federico, Descartes on Polyhedra: A Study of the "De Solidorum Elementis," Sources in the History of Mathematics and Physical Sciences, vol. 4 (New York: Springer Verlag, 1982), 30-32. From Beeckman's journal entries of late 1628, it is clear that Descartes was still using the older cossic notation (it is, of course, conceivable that Descartes used cossic formulas to make things easier for Beeckman) and was solving what we consider to be algebraic problems by geometrical methods close in spirit to those of the Regulae. For more general considerations on the relation between algebra and geometry in Descartes, see Timothy Lenoir, "Descartes and the Geometrization of Thought: The Methodological Background of Descartes' 'Géométrie,'" Historia Mathematica 6 (1979): 355-379; and Pierre Costabel, "La mathématique de Descartes antérieure à la 'Géométrie,'" in Démarches originales de Descartes savant (Paris: J. Vrin, 1982), 27-37.

vide, I claim, is a supporting theory of the nature of human cognition that explains why and how imagination can play the central role in human knowing.

A. THE AIM OF MATHETICAL KNOWING

Let us remind ourselves of the ambition for imagination in the period 1619—1621: corporeal imagination would use figures to image bodies, while intellectual imagination would use bodies to image spiritual truths, the two standing in an analogical relationship to one another. What happens in the Regulae, indeed in the very heart of Rule 4, is that the direction of ingenium takes ultimate aim at human wisdom, the origins of which are the first seeds of truth that 4-B says led the ancients to prefer honesty to utility and virtue to pleasure and, more generally, to acknowledge true ideas of mathesis (disciplina) and philosophy. It is the native endowment of human beings that makes ail this possible, and what Descartes seems to be offering in this work, which intends to sum up what he has learned before he goes on to profounder studies (AT X 379), is a distillation of the simple truths governing all cognition into a small number of rules that henceforth can be used by anyone to judge of the truth in any matter whatsoever. This doctrine governing knowing is based on the premise of the intuitus of simple things and the deductio from these of truths that cannot be known immediately: everything that can be known leads back ultimately to the distinct and easy concept of a pure and attentive mind, and what cannot be so led back cannot properly be known.

This way of knowing is truly universal in scope, since even just the rudiments of it led the ancients to true ideas of mathesis and philosophy (Philosophiae etiam & Matheseos veras ideas agnoverint ; AT X 376 11. 18—19).^[4] What this means is that the first seeds of knowledge enable us, if we look with a pure and attentive mind, to recognize certain objects, like what is virtuous and what is honest, and to see how they stand with respect to other objects, like pleasure and utility.^[5] But immediately after noting these fundamental

[4] This is the only occurrence of 'idea' before Rule 12, and the only one in the Regulae where the word seems to mean something other than "corporeal image." This is a small piece of evidence that, contrary to the Weber thesis, Descartes may have worked on 4-B much later than 1619.

[5] This implies that the recognition of terms is prior to the recognition of propositional truths. Despite the fact that the Regulae frequently relates its teachings to the cognition of propositiones, the basis of the proposition is the mental grasp of a relatively simple object. What words go with any particular grasp is indifferent, but as soon as one compares one thing with another, one arrives at a judgment, which is the basis for all verbal propositions. One must recall again Descartes's penchant for looking to the original meaning of terms: 'propositio' ultimately derives from what is proposed to (propositus ), put before, the mind; as any Scholastic thinker would have noted, the linguistic proposition ultimately derives from the receptivity and acts of the mind.

principles understood mathetically by the ancients, Descartes turns the discussion toward a mathesis conceived along mathematical lines. Whatever else they might have been, Pappus and Diophantus were recognized, not chiefly as wise men, first philosophers, or teachers of ethics, but as mathematicians. And even though order and measure are so broad as to encompass more than strictly mathematical things, the context of the last part of 4-B is dominated by the presence of the traditionally mathematical disciplines. It is quite understandable that most people have interpreted the Regulae as applying a kind of conventionally conceived higher mathematics to all problems. It is important to recognize, however, that in the fundamental intention of the work and in the very notion of mathesis something even higher is implied that the work itself at crucial points falls short of—especially as the universal basis laid down in the first part is gradually steered toward becoming a problem-solving technique.

Yet even at its most technical, the Regulae stays in close contact with cognitive imagination; in fact, it is precisely where the doctrine becomes most mathematical, in the last of the extant rules, that Descartes insists on using images formed both in phantasia and "on paper" (in the external world) to assist the intellect, which otherwise is in danger of error through overlooking relevant considerations. We should recall once more the two imaginations of the "Cogitationes privatae": the corporeal one uses figures to conceive bodies, and the intellectual one uses bodies to figure spiritual truths. The Regulae gives primacy to the corporeal imagination and conceives the intellect as for the most part dependent on mathetical images, yet essentially different from the corporeal imagination because it, the intellect, is spiritual.^[6] The independence of faculties as conceived by the Scholastics is displaced by a more unitary notion of mind. Properly speaking, according to Rule 12, it is now the vis cognoscens or, in a broad sense, intellect that uses figures to conceive things even at the lower level; the operation of the vis cognoscens in phantasia is what imagination in the precise sense is. And, as will be remarked in Rule 8 and reiterated thereafter, body or the corporeal comes into consideration not as existing in the world

[6] One of the "Cogitationes privatae" sheds light on what differentiates this from the earliest position. "For God to separate light from darkness, in Genesis is to separate the good angels from the bad, because a privation cannot be separated from [its] habit: for which reason it cannot be understood literally. Pure intelligence is God" (AT X 218; the order of the concluding sentence, "Intelligentia pura est Deus," decidedly identifies pure intelligence with God). The implication is twofold: the image used in Genesis conveys something of the truth, but it requires analogical understanding; moreover, since only God is pure intelligence, human beings are not, and so the highest truths must in some way be accommodated to their lesser capacities. The intellect of the Regulae seems in many passages to be a pure form of spirit, but its continuing dependence on images and its involvement with the body suggest there is at least a remnant of the impure analogical intelligence described in the Genesis note.

but only insofar as it is conceived, or touched, by the mind. This is the realm of pure phantasms.

What we are likely witnessing in the Regulae is a transformation from the earliest conception of imagination, its aspects, its powers, and its objects, a transformation that is probably implicit in the logic of any cognitive method that prominently features imagination. The notes of 1619-1621 reflected a native realism about bodies: there was no sign that they were being thought of with the proviso "bodies only insofar as they are conceived by the mind," and in fact Descartes presumed to know that there was one active power in things, and that in bodies this was corporeal form. The forms of things in the world were somehow grasped by us, and those forms in turn could be used to grasp higher things. Imagination was the power that enabled us to get a tighter grasp than mere gawking would; it allowed us to draw back from the immediacy of the given so that we might use figuration to better conceive things. Yet this distance loosened the connection between sensation and knowing, it opened up a space where the imagination could function but simultaneously introduced the possibility of going astray because of chance, arbitrariness, and the ineffective use of this power. The imagination is in need of direction, and what pro-rides that direction are the seeds of knowledge given to every ingenium and the small number of rules it is possible to prescribe therefrom. As the private cogitation on Schenckel's art of memory suggested, two ways of approaching the truth open up: proceeding from phantasms to cause and back and forming a phantasm common to each of a group of phantasms, that is, finding the idea or eidos of the group (AT X 230). Either of these ways, or both taken together, would provide guidance for the otherwise unstable, labile imagination. The first method is a way that leads to the real and the causes of things; the second is a way that can proceed along either real or fictive lines so long as order is maintained. But this quite naturally opens up a new problem, for how can we know that a method allowing fictions will ultimately lead to the real and true? Moreover, what is the nature of the real, of the cause, that produces phantasms? Is it itself something that shares in the nature of the phantasm, or is it external to that realm? Does the activity of the mind parallel the activity of the world, or are mind's activity and mind itself something other than the world?

The Regulae can be understood, I think, as an attempt to work out these problems and questions, an attempt that still maintains a basic confidence in the powers of imagination. But certain qualifications and limitations have inevitably worked their way in. Intellect, which in the early notes exercised the higher kind of imagination that is proper to the poets, namely, using corporeal things to figure spiritual things, is now the agent proper in all cognitively relevant acts, from sensation to pure intellection. Bodies are no longer considered per se but as they are touched by the mind.

Although Descartes credits the ancients with recognizing philosophical and ethical truths through the rudiments of mathesis, there is otherwise no mention of how the intellect uses bodies, or ideas of bodies, to figure spiritual—both moral and intellectual—truths. If the primary motif of the early notes was cognitive and spiritual ascent by way of both corporeal and intellectual imagination, the dominant motif of the Regulae is the intellect working on problems that are "beneath' it. It is of course true that even in the early notebooks Descartes was a mathematical and physical problem solver, but then he stood in the first glow of enthusiasm for a technique that promised to carry its practitioner to the highest heights, to which otherwise only the poets easily ascended; whereas now that the technique is being worked out in detail, the heights may be appearing in the guise of the simple, that is, simple objects simply grasped by a pure and attentive mind. The high imaginative leaps of the poet are being implicitly condemned as unmethodical and replaced with the step-by-step ascent of the order-loving philosopher.

B. SERIATION AND THE NATURALNESS OF SIMPLICITY

Rule 4 announced that every examination of order and method properly refers to universal mathesis; Rule 5 announces that the whole method consists in proper ordering and that the two rules that follow will clarify how this order is to be discovered and how it is possible to avoid error. The emphasis on order is not at all surprising, since it is an elaboration of what is implicit in the doctrines of intuitus and deductio. If we are to use these properly (that is to say, if we are to proceed methodically) we must know when they are in order. We cannot inspect or intuit just anything: the problem is that most people are trying to understand a blooming, buzzing confusion, the everyday world, whereas inspection is possible only where we have separated out the confusions and have arrived at something easy; and only once we have recognized these easy things can we reverse direction and begin deductio, proceeding from the simplest to the most complex. This orderly procedure is what Rules 6 and 7 teach. But, as one would expect from what has preceded, this orderly procedure should involve learning both how to put the mind into a pure and attentive state and what the mind is recognizing when it is in that state: that is, there are two subjects under consideration, the activities and powers of mind and the proper objects of those activities. Rule 6 in fact emphasizes the proper object of cognition, under the rubric of simplicity, which makes that object well adapted to the limits of ingenium.

The heading of Rule 6 reads: "In order to distinguish the simplest from involved things and to follow through in order, it is required to observe, in

every series of things in which we directly deduce [or lead down] certain truths one from another, what is maximally simple and in what way all the others are more, or less, or equally, removed from it." The commentary begins by saying that this rule "contains the chief secret of the art, nor is any other more useful in this whole treatise." The reason is expressed in the next clause.

For it advises that all things can be disposed [i.e., set out] according to certain series, not indeed insofar as they are referred to a certain genus of entity, as philosophers divide these things into their categories, but insofar as some can be known from others, such that as often as some difficulty occurs we can immediately notice whether certain ones are prior to others, and which ones, and in what order to survey [them]. (AT X 381)

The secret to the art is thus the disposal of things into series according to a cognitive order of dependency. What is rejected is Aristotelian-Scholastic practice: identifying things according to genus and differentiating character (which two make up the identification of the species) and classifying all the ways in which things exist according to the categories of substance, quality, quantity, place, relation, and so on, that is, according to the order of being rather than of knowing. In addition, as will become clear from the sequel, Descartes is also rejecting the resort to abstract universals, the "terms" of the dialecticians and logicians, in favor of a special kind of particularity. It appears, then, that the secret of mathesis is the proper ordering of certain kinds of things; universal mathesis is the science of the proper order of what is known.

But how can cognition and the cogitation that arrives at cognition exhibit proper order? The answer is implicit in what has preceded. Simple objects are the things most easily knowable of all, and as such they are the origin of knowing: they are first in the order of knowing. One can call them the origin of knowing in two senses. First, there is nothing more knowable than simples; they are perfectly tailored to the nature and capacity of the human mind. Second, from simples we can arrive at other, more complex truths; the place of complex things in the order of knowing depends on how remote they are from the simple firsts. How do we know that we have hit upon a simple? By the fact that it is so easy and distinct that it leaves no doubt in a pure and attentive mind. That is all well and good, but it does not give any guidance about how to put our minds in such a state or any criterion for distinguishing a complex from a simple thing. How do we manage to avoid the aimless wanderings that most of our predecessors have fallen into among the welter of a complicated world?

In the very next paragraph Descartes begins to address this question. "In order however that this might come about rightly, all things, in the sense according to which they can be useful to our proposal, where we do

not look at the solitary natures of them, but rather compare them to one another so that some might be known from others, can be called either absolute or respective." He proceeds to define what he means by 'absolute': "I call absolute whatever contains in itself the pure and simple nature which is in question, like everything that is considered as independent, cause, simple, universal, one, equal, similar, direct [or straight], or other things of this kind; and I call this first thing the most simple and most easy inasmuch as we use it in resolving questions." The 'respective', in turn,

is what participates [in] the very same nature, or at least something from it [= from the nature in question], according to which it [= the respective thing] can be referred to the absolute, and deduced from it [= the absolute] through a certain series; but beyond this it involves certain other things in its conception, which I call respects: such things are whatever is called dependent, effect, composite, particular, many, unequal, dissimilar, oblique, etc. Which respective things are the more removed from absolute things by this, that they contain more respects of this kind subordinated one to another; all which things, we will be advised by this rule, are to be distinguished, and the natural nexus of these among themselves and the natural order are so to be observed that from the last we might by transiting all the others reach to that which is maximally absolute. (AT X 382)

The crucial notions here are a nature, one or more things that contain or participate (in) this nature, and an act of comparison. Comparison is of course implicit in a way of cogitation according to resemblance, whether in comparing images to bodies, bodies to higher things, or any orderable or measurable things to others of the same kind. Comparison operates according to biplanarity. Either one compares two things alongside one another (in the first plane) with respect to some characteristic (which defines the second plane) or one thing is seen with respect to the characteristic.

It is probably not possible to give a seamless, totally consistent account of the absolute, the respective, the simple, and natures, and it is likely that one of the chief causes of Descartes's leaving the Regulae unfinished was the intrinsic difficulty of explaining them. Yet a careful examination of Rule 6 in comparison with other parts of the Regulae allows us to illuminate the key elements.

Descartes's account of the absolute and the respective is not in the first instance about natures (and certainly not about essences) but about what contains or participates them. One has a thing in mind, and one views it in terms of a nature, "the pure and simple nature that is in question." For example, a king might be viewed in terms of independence. It would seem right to affirm that a king, if anyone, is independent. But the method of Rule 6 requires comparison. So now one takes into view not just a king but

also an earl, a knight, a yeoman, and a serf. The resulting relationships to independence have become more complex and articulated. The serf participates least in independence, because he seems close to a slave; one must beware of simply asserting that he participates to degree zero, however, precisely because we have now conceived of another individual who participates in less than he. (Furthermore, one might go on to compare a domestic slave, a slave who works in the stables, another who works in the fields, and a fourth who labors in a mine). The yeoman clearly has more independence than the serf, the knight more than the yeoman, the earl more than the knight, up to the king, who, if his power is not qualified by anything or anyone higher, not merely participates in the nature but wholly contains it. In this series the king is absolute—and that is true even if he is not fully independent in an "absolute" sense. Absoluteness as Descartes defines it is a property of standing at the head of a series with respect to a nature by virtue of wholly containing it or perhaps just participaring it more than any other member of the series.

A fine line etched in a flat marble slab, a railroad track crossing a plain, the path of a beam of light, a bricklayer's plumb line: all participate to varying degrees in straightness, extension, and length. Perhaps none wholly contains the nature straightness (each probably deviates from perfect straightness to some degree); each seems to contain wholly extension in length, and for that matter extension in the other two dimensions as well (the etched line may be just 0.01 mm in width and 0.005 mm in depth, but that is still existence in three dimensions). As we saw in chapter 3, Sec. F (with reference to AT X 453), Descartes conceives actual images as having the three spatial dimensions of length, width, and depth. That is, even a line drawn on paper or a line imagined in phantasia has three-dimensionality (see Rule 14, esp. AT X 446). But the power vis cognoscens is such that we can reliably focus on just one aspect, one nature, at a time.

It is not properly the natures that are absolute or respective but the "things" (res) that contain or participate a nature, or, as Descartes qualifies further in the definition of 'respective', participate "at least something from the nature" according to which it can be referred and compared to the absolute thing that stands at the head of the series and perhaps even wholly contains the nature. Thus it is contained and participating natures that are the foundation of comparison: the absolute things that wholly contain the natures serve as standards from which the respective things are judged to be more or less removed or distant. The respective things, in their turn, can be compared to one another as more, or less, or equally participating a nature. Each group of things can consequently be put into a series—serialized or seriated—according to this comparison in light of their differential participation of a specified nature. Moreover, the absolute

in each such series is the simplest and easiest insofar as it facilitates the solution of problems.^[7]

'Absolute' and 'respective' are therefore pragmatic rather than ontological terms, that is, relative to the practice of comparison for the sake of problem solving. At least for the time being, Descartes is not addressing the question of whether all things can be ontologically reduced to a few simple natures (like thinking and extension). It is not even clear that he is allotting ontological or even epistemological primacy to 'independence', 'cause', 'simplicity', 'universality', 'oneness', 'equality', 'similarity', 'straightness', and the anonymous 'others of this sort'. The passage that privileges these categories in determining the absolute can be read as saying not that these are absolutes but that they are marks or criteria that a thing exhibiting a relation to a nature (other than these) actually contains the nature and thus can be called 'absolute'. For example, the rigid stick and the pair of scales adduced at the end of Rule 9 simply contain, respectively, "immediately conducted natural power" and "natural power simultaneously producing contrary effects"; the former contains unity and cause in a simple manner, the latter cause and equality (of opposites). This interpretation is consistent with Descartes's initial emphasis on the things rather than on the natures the things exhibit. Thus because a king is the cause of the authority of his vassals, he may be called the "most simple and easiest" thing in the series with respect to authority. In comparison to the king's prime minister, the king's authority is undivided (unitary or one); again this is a mark that the king stands in the absolute position.

These reflections lead to a question: is it the natures that are simple and pure, or the things exhibiting them? Recall that in the definition of intuitus it was the grasp or concept that was characterized as easy and the mind as pure. Both the thing and the nature need to be grasped; perhaps the nature is nothing more than a determinate way of grasping (or a determinate focus of attention to) a thing. It is possible that both natures and the things that participate them can be simple and pure, although in the development of the theory of natures that takes place later in Rule 6, and even more in Rules 8 and 12, purity and simplicity come to be seen as

[7] This last characterization in terms of a thing's problem-solving efficacy pushes the serializer to find something absolute not just for a particular series but for all series of the same kind. Thus a wooden ruler might serve as the "absolute" meter for one type of measurement (say, the layout of a practice track), whereas a standard meter made of precious metal preserved in a vacuum at constant temperature would supersede the wooden ruler in an amplified series and be even more effective for setting the standard of scientific measurement (though obviously one uses it to measure only occasionally, for example to establish standard meter lengths for the bureaus of standards of different countries). Most effective of all might be a new absolute defined in terms of a certain number of wavelengths of a certain kind of atom (which is how the standard meter is currently defined).

most properly ascribed to natures. The latter two rules much more directly address ontological questions about natures and manifest multiple tensions and strains. As I shall show, the natures led Descartes away from the resemblance doctrine of his early years; initially conceived as a support for that doctrine, the natures gradually undermined the notion of cognition by resemblance and analogy when the natures were conceived in a radically foundationalist way.

In Rule 6 the place where the development of the account of natures begins is not so much in the definition of 'absolute' as in that of 'respective'. A thing is respective not just when it participates a nature in less than the absolute way but also by involving respects that are connected to one another in a hierarchical way. Besides the participating thing, there is "whatever is called dependent, effect, composite, particular, many, unequal, dissimilar, oblique, etc." This list is in perfect parallel with the previous paragraph's listing of the characteristics that make a thing absolute; each term is the privative or negative correlate of the corresponding member of the first list. The logic appears to be this: A thing we call dependent, or effect, or composite, and so on, is to be understood ultimately as being in relation to the positive form of the characteristic, that is, the independent, the cause, the simple. A thing that in its being grasped (in suo conceptu )^[8] involves dependence, say, a king's vassal, is at a certain remove from the king. If we call to mind that the vassal in turn has his own vassals, we view him not simply in direct regard of independence but, for instance, also as the cause of the powers of his vassals, and this might remind us that his being a vassal toward the king means he is an effect as well.

Descartes emphasizes that 'absolute' and 'relative' are not themselves absolute terms.

And so the secret of the whole art consists in this, that in everything we diligently direct ourselves to that which is maximally absolute. For some things are, under one consideration, more absolute than others, but looked at otherwise they are more respective: as the universal is more absolute than the particular, because it has a simpler nature, but all the same it can be called more respective, because that it exists depends on individuals, etc. (AT X 382)

Moreover, he goes on, things are more or less absolute depending on where they stand: thus if we are looking to (respiciamus ) individuals the species is absolute, but if we are looking to the genus it is respective; likewise

among measurable things extension is something absolute, but among extensions it is length, etc. And finally, so that it might be better understood

[8] This is the fourth-declension noun conceptus, not the second-declension conceptum ; the former is the act of conceiving, collecting, or gathering (here, in the mind).

that we are looking here at the series of things to be known, and not the nature of each one, we have deliberately counted cause and equal among the absolute, although their nature is really respective: for even among Philosophers cause and effect are correlative; here, however, if we inquire which is the effect, it is necessary to know the cause first, and not the contrary. Equals also correspond one to another, but things that are unequal we do not recognize except by comparison to equals, and not the contrary, etc. (AT X 382-383)

These qualifications are baffling unless we keep constantly before our minds—precisely as Descartes admonishes us—that the end is not to understand natures per se, much less things as having unique and essential natures, but to know things according to series, as they exhibit a certain nature or look from the perspective of the inquiry.

'Nature' is in effect used in a threefold sense in Rule 6. First, it is synonymous with Aristotelian-Scholastic 'essence'; but Rule 6 at the outset discounts the value of pursuing natures in this sense, although it does not go so far as to deny that such natures exist. Descartes's assertion that the natures 'cause' and 'equal' are really respective suggests that he attributes a certain reality to at least some natures. Second, it means an aspect or characteristic in view of which a thing can be determinately viewed and grasped, the sense that is most basic to the method explained by Rule 6. Third, it means a restricted class of natures in the second sense, the few natures that are knowable in independence from all other natures, as the second annotation to the rule explains.

It is to be noted second that, strictly speaking, few are the pure and simple natures, the ones it is granted to intuit first and per se, not dependent on any others but either in the experiences themselves or by a certain light situated in us; and these things, we say, are to be diligently attended to: for they are the same ones that in every single series we call maximally simple. All the others, however, can be perceived in no other way than if they are led down [deducantur] out of these, and this either immediately and proximately, or not unless through two or three or more different conclusions; the number of which also is to be noted, so that we might recognize whether these are removed from the first and maximally simple proposition by many or by fewer degrees. Indeed of such a nature is the contexture of consequences everywhere, out of which are born those series of things to be inquired about, to which [series] every question is to be brought back [reducenda], so that it might be examined by a sure method. Because, however, it is not easy to survey all together, and furthermore because they are not so much to be retained by memory as distinguished by a certain acumen of ingenium, some [means] is to be inquired after of forming the ingenia so that they [= ingenia] immediately heed them, as often as there will be need; I have experienced that for this there is really nothing apter than if we become accus-

tomed to reflect with a certain sagacity on each minimum thing out of those that we have already perceived previously. (AT X 383-384)

It would be premature to assume that Descartes had in mind with this third sense of 'nature' the reduction of all things to either extension or thinking, although that is a direction in which it might be taken. How few are the natures of this kind? Descartes does not say, but only our knowledge of the later Descartes will make us assume they are just two; the tenor of the passage suggests a relatively but not minimally small number. Moreover, as we shall see shortly, Rules 8 and 12 allow at least three types of natures, those that are corporeal, those that are intellectual or spiritual, and those that can be either (i.e., exhibited by either corporeal or intellectual things).

What Descartes seems to be aiming at here is building in the methodical investigator a broad and deep experience of natures as points of reference, so that when a problem appears she will be provided with every possible way of approaching it under diverse considerations (they are therefore like topics in the classical, rhetorical sense). We are given assurance that there are simple and pure natures in an ultimate sense, but we do not necessarily have to resort to them in addressing particular problems (though they have the advantage that they are to be found as maximally simple in every—presumably complete—series we might encounter).^[9] A familiarity with such basic natures and their interconnection with others will keep a problem solver from being stumped if she cannot solve a problem expressed in its original terms. For example, in an example provided by Rule 9,^[10] if one is trying to determine at what speed light travels one might recall that it is a natural power, and this in turn might induce one to think of simple, concrete examples of the propagation of natural forces, some temporal, others instantaneous. Here the pure and simple nature, or at least the surrogate for it, is 'natural power', which certainly does constitute a perspective under which many series can be perceived.

But has not Descartes changed the way in which problems are to be solved by shifting from seeing things in the perspective of natures to seeing natures in the perspective of other natures? Does this not displace thinking in terms of concrete things with thinking in terms of the terms? The answer is that a shift may be under way but it is not accomplished. Rather, the Regulae appears to be at the threshold of such a shift. The

[9] Clearly Descartes means not that any one or two of these natures appear in every series, but that, given any arbitrary series, the way to at least one nature will glimmer in the distance (in the last analysis, so to speak).

[10] In the discussion of perspicacity, which is the habit of an ingenium practiced in the division of things into parts and their seriation.

interconnection of natures and the existence of some that are either experienced by themselves or known by themselves through a certain light of nature is not the center of gravity of the method of the Regulae . That center is instead the comparison of things according to their degrees of resemblance to one another, with the principle of resemblance taking the name 'nature'. The method of noting resemblances does not depend on an ontological reduction to fundamental natures. So, for example, one is not limited to using the method of figuration only for solving corporeal problems, because it is the proportional interconnection of things and natures that is at issue, and the concrete, proportionalizing figurations that the Regulae employs are perfectly suited to representing such interconnections, that is, the proportional involvement and intrication of natures, whatever those natures might be. To conceive the Regulae as counseling the direct, purely intellectual thinking of natures would be to fall back into the Scholastic methods that Descartes has rejected.

Later, when in the first half of Rule 12 Descartes suggests representing colors by geometrical figures, he is not effecting the ontological reduction of color to extension (or to motion and extension) but implying that colors are invariably involved with the nature extension. Color itself is a nature in the sense of the Regulae —for example, natural objects participate in colors, and one can serialize objects according to their hues. Although the conveyance of the impression of color to the senses and the mind involves extension, color is not simply replaceable by extension or by a mode of it, since, as Descartes notes, someone blind from birth cannot know color. What he does hope to establish in Rule 12, however, is that the variety of geometrical figures is ample enough so that the patterns of the interrelationships that hold among colors can be accurately represented by the pattern of differences between the representing figures.

What Descartes's method in the Regulae proposes is not to come at concepts abstractly, as is the common Scholastic practice in the quaestio, wherein we would define and distinguish so that we might deduce, but rather to think of the individuals, the instantiations that participate wholly or in part in these natures, and to arrive at an intuitus of the nature by relating (distinguishing and assimilating) individuals to one another in a series. Descartes's method is thus directed toward the classification of individuals, or, to put it more accurately and less misleadingly, toward the viewing of individuals under particular aspects. 'Individuals' here can be taken in a broad sense, limited only to whatever directly appears to the mind as an object. Under this definition it would not be the case that the simple natures themselves are individuals, since they are perceived only as participated in by individuals. Descartes's understanding of the act of the mind, grounded in comparison, or rather seeing a thing along an axis ultimately pointing toward maximally simple natures, involves not irreducible

elements but the grasp (concept) of a thing in light of an aspect it participates or contains. The simplest proposition is thus derived by grasping a thing with a pure and attentive mind in light of a purified aspect or nature; this kind of grasp is intuitus of the most elementary (though not to say elemental) kind.^[11] Intuitus takes place at more developed levels as well, of course, since every deductio is ultimately based in intuitus; simple intuitus is seeing something in its most unelaborated form, simply as it most fundamentally shows itself easily and distinctly within the network of appearances. The object need not be ontologically simple, only phenom-enologically: it must stand out sharply, by virtue of careful preparation (using series), from its aspectual background. Therefore some people might well be able to grasp the Pythagorean theorem in a single intuitus rather than in a complex series of aspects reduced to other aspects.

As has already been remarked several times, we must take Descartes's admonition about his use of Latin words seriously. It should cause us to hesitate whenever we are inclined to take terms in a conventional way. So, for example, the word conceptum, which appears in both definitions of intuitus, should not without further ado be assimilated to 'concept'. Concipere, of which conceptum is the past participle, means most basically 'to take hold of'; as I have already pointed out, it is used in both the Compendium musicae and the Regulae (Rule 12) as a synonym of imaginari. Since intuitus is presented as an act of the mind, the primary sense of the definition should be something like "the hold that the pure and attentive mind has taken on its object." Similarly, to stand in a series is to be proposed (propositum ) to the mind's view in a certain way, and thus every element in the series, including the simplest, is a propositio, and underlying every proposition as statement is a propositio as act of the mind.

What Descartes presents here as a method of inquiry is thus grounded in the very structure and nature of the mind's acts. When the mind presents itself, or is presented with, an individual object, it already has it in its grasp along an axis. Only if we are faced literally with blooming, buzzing confusion do we fall short of this state. But usually we are already presented with more: the identification of this individual as shaped thus and so, of a certain color, of a certain kind. That is, we are presented with various aspects or axes along which to see the thing, some quite definite, others rather vague. If it is definite, then we have already (implicitly) placed

[11] Further light is shed on what is involved in the simple intuitus of natures by Rule 12's account of universals. We are told there that 'limit' is not properly a nature, even though it seems to be more general than terms like 'shape', because it is conflated out of different realms, for example, limit with respect to space and limit with respect to time (AT X 418-419). This means that the ultimate criterion of whether one is dealing with a real (much less simple) nature is whether it is unequivocally instantiated in things and also that not every universal is a nature. See also footnote 36, below.

the individual in a specific location along the relevant axis; if vague, then we can, by applying industry, seek a precise location along the axis. The axis and the indefiniteness, or even the axis and unexplicitated definiteness, present us with the task of precise location: that is, we are faced with something to inquire after, a quaestio, which can be resolved if we identify sufficient determining factors and also represent to ourselves, if need be by the method of geometrical instantiation, the order and the definite proportions that hold between those factors, the natures or aspects.

C. SERIATION AND GEOMETRIC INSTANTIATION

The evocation of the least things in our past experience at the end of the second annotation to Rule 6 leads quite naturally to the third annotation, which sounds a recurrent theme of the Regulae by inviting us to practice our powers first not with complex but with fairly simple matters.

Third and finally, it is to be noted that the beginning of studies is not to be made with the investigation of more difficult things; but, before we gird ourselves for certain determinate questions it is needful previously to collect spontaneously, without any choosing, obvious truths, and tentatively thereafter to see whether certain others can be deduced from them, and in turn others from these, and so forth. This having been done, one must attentively reflect on the discovered truths, and must diligently think why we were able to find some earlier and more easily than others, and which ones these are; so that from this we will judge, when we set upon some determinate question, which other things to be discovered it will help to attend to first. (AT X 384)

The first annotation brought our attention to natures and the degree to which things participate in them; the second points out that these differences in participation give rise to series leading up from the given objects to the maximally simple natures and back; this third annotation finally gives us an easy, "bootstrapping" method for familiarizing ourselves with degrees of participation in series by advising us to turn first to the easiest kinds of problems. This is accessible to anyone and everyone who possesses reason in the least degree, regardless of previous experience. You do not even need to exercise prudent choice; just take the first obvious truth that comes by and see where it leads, then look back over the whole sequence of one thing's being related to another to gain insight into how problems are solved. This provides not only knowledge of the form of argument but, even more important, the material out of which problems are made. After all, in familiarizing ourselves with the dependency between certain truths and others we are acquainting ourselves with series that lead back or point to natures. Any particular practice exercise may not necessarily lead us all the way back to maximally simple natures, but it is these

to which the whole method refers and on which its efficacy depends. In the last analysis these practice exercises teach us how to maneuver within the cognitive realm laid out on their basis.

The example Rule 6 gives of choosing a simple truth and seeing where it leads is the relations between the members of the series of numbers 3, 6, 12, 24, 48, and so on. We notice, for example, that 6 is double 3 and inquire what the double of 6 is, namely, 12, and so on; from this we deduce that the proportion between each successive pairing (between 3 and 6, between 6 and 12, between 12 and 24) is the same and thus that the whole series constitutes a continuing proportion. Descartes grants that this seems a bit childish, but that is simply because these things are consummately perspicuous. "By reflecting attentively on this I understand with what reason [or ratio, ratione ] all questions that can be proposed about the proportions or habitudes of things are involved, and in what order they ought to be inquired after: which one thing embraces the sum of the whole science of pure Mathematics [totius scientiae purae Mathematicae summam ]" (AT X 384-385). This is a remarkable claim, that all of pure mathematics derives from these simple reflections; it evinces the power and ambition of the proportionalizing method he is proposing.

The rest of the rule explores further what is involved in this simple numerical series and shows how the ease or difficulty of discovering the different members and their relations to one another depends on the particular way in which a problem is posed. For example, given 3 and 6 it is easy to find 12, but if you were given 3 and 12 and asked to find the mean proportional (the number between 3 and 12 such that 3/x = x /12) the problem would be more difficult; and even more difficult would be to find two mean proportionals between 3 and 24 (numbers x and y such that 3/x = x/y = y /24). Of course, since we already have the series in hand, these are artificial rather than real problems, but if we change the given numbers to, say, l0 and 150, we recognize that it is not at all obvious what the mean proportionals are and that it would take some at least minimal intellectual agility to find them. (The formulas I have given in parentheses in the previous two sentences greatly facilitate the exact solution of the problem; such formulas are not given in the Regulae, and one might reflect that our ability to reduce the problems to such simple equations is precisely the sort of thing that Descartes was trying to cultivate.) But if we conclude that just adding to the number of proportionals to be found makes the problem increasingly difficult, we will be wrong; for if we are asked to find three mean proportionals, we might notice that this problem can be reduced to simpler ones: namely, first find the mean proportional between the two extremes, say 3 and 48 (this is 12, since 3/12=12/48), then find the mean proportionals between this number and each of the extremes (6 is the mean proportional between 3 and 12, 24 is the mean between 12 and 48).

These discoveries about this particular series of numbers depend on the fact that each member of the series is derived from 3 by successive doubling; the series is 3, 2 × 3, 2 × (2 × 3), 2 × (2 × 2 × 3), and so on, or, using algebraic notation and a generalized formula, the nth member of this series is given by the formula 2^n-1 × 3 (again, Descartes does not give such a formula here). Putting this into the framework of the entire Rule 6, we can say the following: in this series, the absolute member is the first, in that it contains wholly and purely the fundamental nature, 3, that gives rise to the whole series, and all the other members are respective, relative to this.^[12] Of course, the series is determined not just by the number 3 but also by the way in which all the other members participate in 3, that is, according to the formula 2^n-1 × 3 (according to the common nature 'double'; the common natures will be explained in Sec. D, below). Thus the secret of the whole art, Rule 6, teaches that the key to solving problems is twofold: (1) finding the ultimate perspective, the nature, involved in a thing; and (2) analyzing the determinate relationship of this thing to similar things. All problems always involve at least one given aspect or nature and the relation of at least one thing to that nature.

Thus Descartes in Rule 6 gives not just a method of problem solving but a philosophical theory of what a problem is. The rule is key to understanding why Descartes insisted that his art or method was different from the Scholastic method predicated on terms and concepts: rather than seek the essence and the attributes of a thing, it fosters noting what is given, remarking similarities, differences, and resemblances, and identifying axes or aspects under which the given can be viewed. Having mastered the simple but powerful mathematics of proportion we can thereafter address any problem and either solve it or come to the realization that with our current resources it is unsolvable.

Indeed, one might consider the entire Regulae as a treatise on what constitutes a problem. The medieval notion of quaestio is redefined: no longer is it a question that has arisen because of controversy, because of a conflict of interpretations, which is resolved by attending to the necessary conceptual distinctions for the matter at hand or the thing in view. In Descartes's eyes these are disputes over words. He urges instead attending to the object presented to the mind and the possible aspects under which it can be grasped, estimating the relation of the object to other objects having the same aspect so that its relative or proportional distance from the pure and whole participation in that aspect can be more or less precisely deter-

[12] Note that the simplest element might not always appear among the given members of the series, for example, if the series had a more complicated formula (though it would not need to be very much more complicated; e.g., the one determined by the formula [2 × 3]-1).

mined, and then determining the interrelations of these objects with respect to different aspects; if a sufficiently well-developed network of relationships is established, all the relevant degrees of interparticipation can be reckoned; if not, either more work remains to be done or the problem is unsolvable.

D. IMAGINABILITY OF SERIATION, IMAGINABILITY OF INTELLECT

Things participate in certain natures to varying degrees, and the human ingenium can easily ascertain whether two things participate equally, or one more and the other less. This ability to compare participation is the psychological foundation of order and, when a unit of comparison can be ascertained, measure. All relationships of order and measure can be simply represented by figures. Where order is in question the figures can be either discrete or continuous (e.g., a collection of three points to represent the third position versus a line segment three units in length); where measure is in question it is often necessary to use a continuous representation, unless all the measures involved are commensurable (i.e., expressible either in whole numbers or in rational fractions).

In Rules 12 through 14, Descartes brings to a focus the use of imagination in cognition. This use observes the powers and limits of human ingenium raised in the first part, and it develops imagination into a universal instrument of knowing.

Rule 12 lays down both a schema of the faculty psychophysiology of cognition and a theory of natures. In accordance with the preliminary discussion in Rule 8, the natures are divided into those that are maximally simple and those that are complex or composite (AT X 399). The simple natures cannot be false; only in natures composed by the intellect can there be falsity. In Rule 8 the maximally simple natures are divided into the spiritual, the corporeal, and those pertaining to either of these; Rule 12 calls them intellectual, or material, or common to both (AT X 419).^[13] But Descartes quickly sets aside the question of how things actually exist in favor of how they appear to the mind.

For if, for example, we consider some extended and figured body, we even say that, from the perspective of the thing, it is something one and simple;

[13] These divisions appear to be, in the language of Rule 7, enumerations that are sufficient rather than complete or distinct. An ontologically curious philosopher would eventually wonder whether the terms 'corporeal', 'intellectual', 'common', and so forth, refer to genuinely simple natures or simply to abstract generalizations, like the term 'limit' (as explained at AT X 418-419). See footnote 11, above.

and indeed, in this sense, it cannot be said to be composed out of corporeal nature, extension, and figure,^[14] because these parts never existed each distinct from the others; but from the perspective of our intellect, we call the composite something [composed] out of these three natures, because we understood the single things separately before we could judge these three in one and discover them simultaneously in the same subject. For this reason, since here we are not treating of things except insofar as they are perceived by the intellect, we call only those simple the cognition of which is so perspicuous and distinct that they cannot be divided into several things more distinctly known by the mind: such are figure, extension, motion, etc.; all the rest, however, we conceive as in some way composed of these. (AT X 418)

Once again a pragmatic epistemological aim rather than an ontological one governs the discussion. It is not so much that the natures are being treated as fictional as that their mode of existence in the thing is allowed to be different from their mode in the mind. That they certainly are con-rained or participated by things is not in question. The principle of the division of natures into intellectual, material, and common classes shows this.

Those things, which with respect to our intellect are called simple, are either purely intellectual, or purely material, or common. Purely intellectual are those that are known by the intellect through a certain inborn light, and without the aid of any corporeal image: for it is certain that there are some such things, nor can any corporeal idea be feigned that would represent to us what cognition is, what doubt, what ignorance, likewise what is the action of will that it is granted to call volition, and similar things; all which we nevertheless truly know, and so easily that for this it suffices that we be participators in reason. Purely material are those which are not known unless they are in bodies: as are figure, extension, motion, etc. Finally, those are to be called 'common' that sometimes are attributed to corporeal things, sometimes to spirits without distinction, like existence, unity, duration, and similar things. To this also are to be referred those common notions which are like a certain chain conjoining other simple natures to one another, and by the evidence of which is supported whatever we conclude by discursive reasoning. Viz., these: those things that are the same as a third are the same as one another; likewise, what [two things] cannot be related in the same way to a third thing also have some difference between them, etc. And even these common ones can be known either by pure intellect, or by the same [intellect] intuiting the images of material things. (AT X 419-420)

Descartes also includes among the simple natures negation and privations of the positive natures. Abstractions like 'limit', which we derive by abstract-

[14] Note how Descartes distinguishes these as though they were three maximally simple natures. In the later philosophy, extension would be the foundation of the other two.

ing from different simple natures (in cases of shape, duration, motion, and the like), are not numbered among those natures.

The crisis in Descartes's thinking posed by his development of the natures doctrine will be taken up in the next two chapters. For now we shall consider only that the natures of Rule 12 are sufficient to ground the universality of imagination as a problem-solving instrument. This is true despite the fact that the intellect recognizes the purely intellectual natures by an innate light without representing them in images.

Of course Descartes leaves no doubt that there can be pure intellectual knowledge, but the examples he gives have some curious features. All of them are actions or passions of the mind or soul: knowledge, doubt, ignorance (a passion), volition. None of the natures involved is a direct object of cognition, however: willing would be recognized as such in the act of willing something else, and similarly for the others. All of these acts can have as their object an image or a body. Moreover, the fact that a certain nature can be recognized by intellect alone does not imply that imagination cannot be of assistance in addressing questions about intellectual things, especially since Descartes's problem solving in the Regulae is not a matter of determining essences. For example, a kind of order should be possible even with respect to knowing and willing: after all, the Regulae itself is an attempt to discover and encourage order in knowing. One might order volitions according to objects. One might ascertain that a certain order or proportion exists between willing and knowing in even purely intellectual acts, and this order and proportion could be represented schematically by discrete or continuous figures. Representing one act of will as participating in understanding more than another, or adjudging that one act of will is more intense than another, does not require ontologically reducing something purely intellectual to corporeal form. That a volition in itself is noncorporeal and as such has no similarity to the images of the corporeal realm is one thing; that my participating in willing is so-and-so intense and relates to another act of will in a certain proportion is quite another, having to do with the interrelations of things and natures. Recall that the common natures include the "common notions" that are like a "chain conjoining other simple natures to one another, and by the evidence of which is supported whatever we conclude by discursive reasoning." All discursive reasoning, even that about intellectual natures, thus has recourse to common natures, which can be instantiated by either intellectual or corporeal things. That there are natures common to both spirit and corporeal things means that there are ways of at least indirectly representing spirit through corporeal symbols. The "existence, unity, duration, and the like" of a spiritual thing can be represented by a parallel and proportionate corporeal instantiation of "existence, unity, duration, and the like."

E. SOLVING PROBLEMS BY PROPORTION

"All human science consists in this one thing, that we distinctly see in what way these simple natures simultaneously concur for the composition of other things" (AT X 427). The common error in encountering any problem is to suppose that there is involved some kind of previously unknown entity; the particular temptation to which the educated succumb is to substitute their learned conceptions and learned vocabularies, which are really unintelligible both to themselves and to others, for what they have experienced.

But whoever cogitates that nothing in [for example] a magnet can be known that does not consist of certain simple natures, known through themselves, is not uncertain about what to do: first to collect diligently all experience that it is possible to have about this stone, out of which then he tries to lead down [or deduce] what mixture of simple natures is necessary for producing those effects that he has experienced in the magnet; once these are discovered, he can boldly assert that he has experienced the true nature of the magnet insofar as it could be discovered by a human being and from the given experiences. (AT X 427)

The key is to ascertain what natures might be involved and how they can be combined, the chief device for which is the determination of orderly relations between things and natures noted by the ingenium's powers of comparison, serialization, and proportionalization.

Rule 12 ends with a remark that prepares the way for the rules of the second part. It says that everything will be divided either into simple propositions or questions (problems).

As for simple propositions, we treat no other precepts than those which prepare the power of knowing [vim cognoscendi]^[15] for intuiting any object whatever more distinctly and scrutinizing it more sagaciously, because these must occur spontaneously, nor can they be inquired after; which we have embraced in the twelve preceding precepts, and in which we believe we have exhibited all things that we judge can render the use of reason somewhat easier. (AT X 428-429)

What involves a simple nature simply cannot be investigated, it is simply seen. But when there is any interconnection or relation not clearly seen there is a question or problem of proportion to be resolved. Like the Scholastics, Descartes proposes to address quaestiones. The logic of addressing them, however, is not dialectical and verbal, that is, dependent on past texts

[15] Note the similarity to, but also difference from, Rule 12's 'vis cognoscens' (for which the form here would be 'vim cognoscentem'). The variations in terminology add further weight to the contention that Descartes's thinking was in flux.

and authorities, but the serializing and proportionalizing logic of the interconnection of natures—mathesis universalis.

Descartes divides questions or problems into two kinds, those that are perfectly understood, to be treated in Rules 13 through 24, and those that are not, to which the projected Rules 25 through 36 were to be dedicated. The reason for this division is pedagogical: the last twelve rules would have presupposed acquaintance with the preceding dozen, and "we teach those earlier with which also we think we should first be occupied in order to cultivate the ingenia" (AT X 429). The perfectly understood problems of Rules 13 through 24 require that we perceive three things: "by what signs that which is being sought can be known, when it presents itself; what it is, precisely, from which we must deduce it; and in what way it is to be proved that these things so depend on each other that one can by no reason [ratio ] be changed, leaving the other unchanged" (AT X 429). This is a case of learning how to discover a conclusion not by deducing one thing from a single simple thing, "but one thing depending on many implicate things simultaneously, evolved so artfully that it requires no greater capacity of ingenium than for making the simplest illation' (429). This is shown most easily by using examples from arithmetic and geometry, which are useful for the practice needed to acquire the technique.

Rules 13 and 14 lay the groundwork for the treatment of questions or problems, and as such they are both an extension of what precedes and an anticipation of what is to come. Descartes begins by distinguishing what he is doing from the practice of Scholastic logicians. We are not to search for the middle term connecting two extremes but rather (in echo of the last paragraph of Rule 12) to recognize that in every problem there is something unknown, that this unknown must be designated in a determinate way, and that this designation must be made in terms of what is already known.

This is precisely what we teach children in algebraic problem solving today. "A truck sets out from city A for city B at 1 P.M. It arrives in city B at 3:30 P.M. The truck averages 45 miles per hour. How far apart are cities A and B?" There is something unknown, the distance between A and B; we designate this by x . This designation has to be put in relation to things that are known, the time oft. ravel (2.5 hours) and the average velocity (45 miles/hour), by means of an equation: x = 2.5 hours × 45 miles/hour = 112.5 miles. This kind of problem reflects the progressive complication that marks the division of the Regulae into three major parts. In comparison with the simplest things—for example, noticing that the truck is moving, that it is moving faster than a car, that cities A and B are different but still both cities—this problem is complicated. But in conformity with Rule 12's criterion (AT X 429), the answer is perfectly knowable in terms

of the given because all the proportions are interrelated in a way that allows a unique solution (in contrast to problems in which the information is inadequate—for instance, what the speed of the truck was at precisely 2:00 P.M.—or the interrelations determine no solution or multiple solutions). In conformity with Rule 8's distinction between the second part, treating of natures deduced from simple and evident natures, and the third, treating of natures that presuppose natures composite in reality (AT X 399), these types of problems involve natures so perspicuously interrelated (space, time, motion) that a further knowledge of natures, one that can be acquired only through the experience of compositions accomplished in reality, is not necessary.^[16]

What seems quite simple to us was not yet a permanent acquisition of mathematics around 1630; Descartes of course was one of the first to insist on the need to name unknowns and to manipulate them mathematically as though they were ordinary numbers. Even this is to put a modern spin on things, for the extant part of the rules does not explicitly deal in equations, and, given the kind of geometric rather than arithmetical representation and manipulation it presents, we cannot simply assert that it is teaching the elementary algebra of problem solving. Rather, it is the elementary art of problem solving, which is the largest part of what Rule 4 calls mathesis universalis.

The remainder of Rule 13 gives instruction in what we should do as we begin to approach a problem, in the initiating heuristic of problem solving. We are told that if the problem is, for instance, about the nature of the magnet, we already possess the meanings of the terms 'nature' and 'magnet' and thus have a preliminary determination of how to proceed. In another case we might be asked what is inferable of the nature of the magnet from the experiments published in William Gilbert's De magnete (1600), whether they be true or false; in yet another we might be given specific data about the size and weighting of strings and asked to determine whatever we can about the nature of sound. Descartes grants that these specific problems are imperfect, in that we cannot be sure at the outset that we have everything needed to solve them, but nevertheless they can give an idea of how we begin a "reduction" to a perfect problem by trying to enumerate the simplest parts or natures. Descartes promises further that in his explanations "it will appear also in what way this rule can be observed, so that the well-understood difficulty be abstracted from every

[16] Such problems of real complexity—presumably Descartes is thinking of natures no less complex than color, which Rule 12 grants can be known only through real-life encounters—require a background experience that goes far beyond what is accessible by a certain inborn light of reason (the distinction is made in this way in Rule 6, AT X 383). Only natures known to us in the latter way axe suitable for the most perspicuous kinds of problems and thus treated in the second part of the Regulae.

superfluous concept, and so far reduced so that we will not think [= cogitate] of busying ourselves more widely with this or that subject but only in general concerning certain magnitudes to be compared/composed^[17] with one another" (AT X 431).

Rule 13 reminds us that Rules 5 and 6 had advised taking difficulties back to simple things, and Rule 7 dividing them, "so that afterwards ail may be comprehended simultaneously by a sufficient enumeration' (AT X 432). If there are many experiments concerning the magnet, I must run through them separately, one after another; if the question is about sound, I will compare first strings A and B, then A and C, and so forth, with the aim of comprehending the enumerated whole. Descartes remarks that one needs to resort to Rules 5 through 7 only, using pure intellect with respect to the terms of each proposition, before going on to Rules 14 through 24, and he promises to explain how this should be done in the never-written third part of the Regulae .

The pure intellect must do its work before the rules explicitly applying spatial figuration can be used effectively. Why? Because only intellect has the power of making the distinctions necessary for the "abstraction [of a question] from ail superfluous concepts,' and distinctions are an exercise of the power of negation.^[18] In the initial formulation of a problem the given data and terms are treated as abstract units of comparison; their specific content is not appropriated until the actual solution process begins in accordance with Rules 14 through 24. Thus, although this distinction making is a work of the intellect alone, it is not the work of an intellect dealing with only noncorporeai objects; quite the contrary. The power to cognize and recognize differences is fundamentally that of intellect, in whatever ideational or imaginal form the differences appear.

Thus it is the intellect that initiates a question by setting up its fundamental terms. Once the question has been set in this manner "it is to be seen precisely in what its difficulty consists, so that this thing, abstracted from everything else, might more easily be solved" (AT X 437). "Here, therefore, we say only this to be worthwhile, to review in order ail those things that are given in the proposition,^[19] rejecting those that we see manifestly do not contribute to the thing [ad rem non facere aperte ], retaining the necessary ones, and sending the doubtful back for a more diligent examination" (AT X 438).

[17] A has componendas, H comparandas ; the difference is not of great importance, as comparison is the way ingenium has of analyzing the composed things it experiences, and it is in accordance with its comparative understanding of series leading to natures that it learns how things are composed.

[18] See the next section, below, for this point as it is raised in Rule 14's discussion of the different ways in which the term 'extension' is to be taken.

[19] Quite clearly, 'proposition' (propositio ) must be understood here as "what is proposed or set forth to the mind."

These are the concluding words of Rule 13. At this point, if the problem has not resolved itself, the work of imagination must begin. As the Rule 14 heading puts it, "this same [question] is to be transferred to the real extension of bodies, and the whole proposed to the imagination through bare figures: for thus it will be perceived far more distinctly by the intellect" (AT X 438).

It would be mistaken to minimize this as counsel that applies only to problems concerning corporeal things, for it is presented not as a special case of problem solving but as the appropriate way of taking further any problem that has not been solved already by the simplifying and distinguishing work of Rule 13. What makes this method of mathesis universal is precisely (1) the proportionality of things' participating in natures and (2) the enumerable interconnections of one nature with others, both of which can be perspicuously represented in bare figures.

This is exactly the point made by the first paragraph of the commentary to Rule 14.

So that, however, we might also use imagination as an aid, it is to be noted that whenever one unknown thing is deduced from some other already known, not for all that is some new genus of entity discovered but rather this whole cognition is only extended thus, that we perceive the thing inquired after to participate thus or so [in] the nature of those which were given in the proposition. For example, if someone is blind from birth, it is not to be hoped that we can ever by any argument bring about that he perceive the true ideas of colors such as we have with sound senses; but if someone has at some time seen the primary colors, although never the intermediate and mixed ones, it can happen that he might form images also of those that he has not seen from the similitude of others through a certain leading-down [deduction]. In the same way, if in the magnet is some genus of entity to which our intellect has hitherto perceived nothing similar, it is not to be hoped that we will ever know that through discursive reasoning; but it will be necessary to be instructed either by some new sense or by the divine mind; but whatever in this matter can be discharged by human ingenium, we shall believe ourselves to have gained, if we perceive most distinctly that mixture of already noted entities or natures that produces the same effect that appears in the magnet. (AT X 438-439)

The following paragraphs explain as concisely as possible that when we know something it happens by means of the proportionalizing comparison of one thing to another, according to their differential participation in a nature or common idea/appearance (except in the case of the intuitus of a single thing). That comparison, to become perspicuous, requires an equalization (setting equal) by proportions—today we understand this as setting up an equation—and all comparison, insofar as it involves differences of degree with respect to a nature, can be brought back to differences in magnitudes that are easily depictable in the imagination.

And indeed all these entities already noted, such as are extension, figure, motion, and the like, enumerating which is not in place here, are known through the same idea in diverse subjects, and we do not imagine the figure of a crown differently if it is silver than if it is gold; and this common idea is transferred from one subject to another in no other way than through simple comparison, through which we affirm that what is sought is, according to this or that, similar, or the same, or equal to some given: so that in all ratiocination we know the truth precisely only through comparison. For example, in this: all A is B, all B is C ; therefore all A is C ; the sought-for and the given are compared to one another, namely, A and C, according to the fact that each is B, etc. But because, as we have already often warned, the forms of syllogisms help not at all in perceiving the truth of things, it will profit the reader if, these things being outright rejected, he conceive all cognition whatever that is not had through the simple and pure intuitus of one solitary thing to be had through the comparison of two or several to one another. And indeed almost the whole industry of human reason consists in this operation's being prepared; for when it is open and simple there is not need of the assistance of art for intuiting the truth which is had thereby but only of the light of nature.
And it is to be noted that comparisons are said to be simple and open only when the sought-for and the given participate [in] some nature equally; all the rest, however, require preparation for no other reason than that the common nature is not equally in each, but according to certain other habitudes [or relations] or proportions in which it is involved; and the preeminent part of human industry is to be located in nothing other than reducing these proportions so that equality between the sought-for and something that is known is seen clearly.
It is to be noted consequently that nothing can be reduced to such equality unless it receive more or less, and that all this is comprehended through the word 'magnitude': so that after the terms of the difficulty are abstracted from every subject according to the preceding rule, we understand ourselves to attend successively only to magnitudes in general. (AT X 439-440)

One can hardly exaggerate the importance of this passage, for it not only illuminates what follows but articulates the foundation of the method in the first twelve rules as well. It is comparison that is essential to knowing, and knowledge by means of comparison is a judgment of either similarity, or sameness (i.e., identity), or equality; and this judgment is made in light of a given that is viewed in a particular way (secundum hoc vel illud, according to this or that). The kind of ordered and proportionalized seriation envisioned in Rule 6 is possible and truthful only on such a basis. Two things that participate in one nature participate in it in a similar, an identical, or an equal degree. If identical, then the two things are either one and the same or at most different instances of the same thing, and they will occupy the same rung in the series; if merely equal, they are different things that nevertheless are on the same rung with respect to the particular nature in

question (with respect to other aspects, however, they will differ); if similar, then they participate in the same nature in different ways or degrees. Often we can easily see that one is more or less participant than another, but this becomes perfected knowledge only once we have ascertained this relation precisely, that is, only once we have discovered the proportion of participation. Whether the participation is equal or proportional, however, this relation participates in quantity, and because of this it can be expressed in magnitudes. This holds true whether we are dealing with a corporeal nature, a mixed or common nature, or a purely intellectual nature. With respect to the relation of participation in natures, there is no essential difference among these types, and even the relations of purely intellectual things can be expressed through corporeal magnitude. This knowledge by proportionalization is truly a mathesis universalis.

Descartes says that all knowledge that requires two or more things is comparative; thus deductio and enumeratio are embraced by this category. The only exception is the simple and pure intuitus of a solitary thing. Yet within the framework of the Regulae it is not clear whether we can ever have such intuitus. The only candidate for it is a nature, or rather a simple nature/thing. Do we ever have intuitus of pure extension? As we shall see shortly, Descartes says in Rule 14 that there cannot be extension without an extended thing. So the question becomes, Can we have intuitus of a particular extended thing? But then we would appear to have at least two things: the particular instantiation and also the simple nature in light of which we are viewing the thing (which, presumably, we could also view in light of figure, motion, or other natures).^[20] This may in fact impart some clarity, retrospectively, to the point of calling the highest thing in a series constituted according to Rule 6 'absolute'. It is absolute by virtue of wholly containing the nature in question, rather than merely participating in it to some degree (see AT X 381 11. 22-23 and 382 11. 3-6). Any extended thing wholly contains the nature extension (there still can be a differentiation based on the quantity of the extension). Even a finely drawn line is extended in three dimensions and thus contains extension wholly. It also participates differentially in linearity or in curvature; the straightest one we are able to draw would serve as the absolute member of the series 'linearity'. In an important sense, the full character of the (simple) nature itself becomes purely and simply apparent only if it is seen in its preeminent instance or instances, and those instances are seen as preeminent only in comparison to instances that participate in the nature to a lesser degree.

[20] Once again, our impressions of the later Descartes mislead us. We have become accustomed to thinking that a pure inspection by the mind of a simple idea in and by itself is not only possible but the very stuff of Descartes's thought, to the point that we tend to overlook evidence to the contrary.

Therefore even the simple nature cannot be seen as simple except in contrast to other things; and without a full-blown innatism, the likes of which the Descartes of the Regulae does not seem willing to countenance,^[21] one would have to rely on something like habit, thus on memory, for a series-less evocation of the simple natures. The only way that one might be able to escape this paradox without abandoning the framework of the Regulae would be to argue that pure and simple intuitus is properly interpreted as the careful and penetrating viewing of a single thing actually present to consciousness in its most crucial or essential aspect.^[22] That perhaps does not fully eliminate the possible taint of plurality, although it would appear to take an important step toward the kind of inspectio mentis that divides the world into two mutually exclusive simple substances, res cogitans and res extensa, and identifies every particular thing as an instance or a modalization of one or the other.

Only one additional point is needed to make the doctrine of the rules that follow almost a mere consequence of the foregoing discussion: the role of imagination, to which Rule 14 immediately turns.

So that indeed we now also imagine something and do not make use of pure intellect but of the help of species depicted in phantasia: it is to be noted finally, that nothing is said about magnitudes in general that cannot be referred also to any in particular.^[23] From which it is easily concluded that it will be not a little helpful if we transfer those things that we understand about magnitudes in general to that species of magnitude that, among all, is drawn most easily and distinctly in our imagination: indeed that this is the real extension of a body abstracted from every thing other than that it is figured follows from what was said at Rule 12, where we conceived phantasia itself with existent ideas in it to be nothing other than a true body, real, extended, and figured. Which thing is also evident per se, since in no other subject are all differences of proportion more distinctly exhibited; for although one thing can be said to be more or less white than another, likewise one sound more or less sharp, and similaxly for other things, we nevertheless cannot exactly define whether such excess consists in a duple or triple, etc., proportion unless through a certain

[21] In supposition six of Rule 12 we are advised that falsity can be avoided by resorting solely to what we have actually experienced (AT X 422-423). The inborn seeds of knowledge that the Regulae counts on are not sufficient to generate a priori all knowable things.

[22] In the Principles (I 52-53, AT VIIIA 25) we know a substance through an attribute only, never in itself. Thus the mature philosophy retains the biplanarity characteristic of the early period.

[23] Although the parallelism of "de magnitudinibus in genere" with "ad quamlibet in specie" suggests this translation, there is also a parallel to "speciebus in phantasia" in the first part of the sentence, so that this could be rendered: "nothing is said about magnitudes in general that cannot also be referred to any in an image." But this does not affect the interpretation.

analogy to the extension of figured body. Therefore let it stay settled and fixed that perfectly determined questions contain scarcely any difficulty beyond that which consists in evolving proportions into equalities; and all that in which precisely such a difficulty is discovered can easily, and must, be separated from every other subject, and then transferred to extension and figures, about which things alone we now will treat up to Rule 25, every other cogitation being omitted. (AT X 440-441)

This passage makes the truly remarkable statement that it is in phantasia that the differences of proportion are more distinctly evident than anywhere else; as we shall see from the next section of Rule 14, this implies as well that intellect is incapable of recognizing proportions in any distinct way without the aid of imagination. Nor, to emphasize it once more, is this procedure being restricted to the treatment of questions about corporeal and corporeal/intellectual natures. It is a perfectly general procedure, for any problem whatsoever. Even the transfer of degree of participation in natures to line lengths is presented as an example of what this procedure allows, not as its essence, since many other kinds of figuration are possible and permissible, depending on the problem.

The doctrine of natures or common ideas, whatever their provenance, is the foundation of the cognitively effective application of phantasia. Degree of participation or containment can, indeed must, always be compared using magnitudes. Since proportions can be maintained at any arbitrary scale, one can take any magnitude one pleases and call it the whole or, in cases of measure, the unit; all the relations of other things to this magnitude can be expressed proportionally using larger or smaller magnitudes. Moreover, once one has led the participation down to magnitudes one does not have to keep in mind the natures themselves. One can solve (equalize) degrees of participation while completely prescinding from the special character of the nature not expressed in magnitudes. Only at the end of the equalization process, when one has determined the magnitude that corresponds to the unknown thing by treating it as though it were known—what we would call "solving equations in unknowns"—does one need to remind oneself what the nature is, just as in physics one can solve equations using techniques of pure mathematics while ignoring the units in question until the end, when one has to remind oneself that they are centimeters, or joules, or kilogram-meters per second squared.

These foundational principles of the imaginative problem-solving technique are elaborated at the end of Rule 14 (after an intervening discussion of different ways in which extension is imagined and understood; see Sec. G, below).

Here therefore we turn to the extended object, considering nothing else in it but the extension itself and abstaining by industry [or on purpose] from the word 'quantity', for there are certain philosophers so subtle that they

have distinguished that also from extension; but we suppose all questions so deduced that nothing else is sought for than a certain extension that is to be known, by its being compared with a certain other known extension. (AT X 447)

This will be accomplished by reducing involved proportions, all expressed extensionally, to a single one that is equal to the unknown; this can be done no matter how many differences of proportion there are. The crux of the matter is figuring out ways of presenting these differences and "reducing" their relationships so they are visible to the eye and the imagination. "And therefore it will be sufficient to our institution if in this extension we consider all those things that can assist the exposition of the differences of proportion, of which there occur only three, namely, dimension, unit, and figure" (AT X 447).

Dimension and unit have to do chiefly with what is measurable; figure, with both the measurable and the orderable. Rule 5 had asserted that the method was a matter of ordering and arranging things, Rule 6 that the participation of things in natures could be used to produce orderly and even measurable series. This part of Rule 14 shows how this order can be imaged and develops an understanding of measure as derivative from a certain type of order. Measuring requires a unit of measure; a unit of measure must measure something, a dimension; a dimension is a particular way of grasping the things in question. It is either a nature or naturelike—even if it turns out that it does not actually exist!

By 'dimension' we understand nothing other than the mode and reason according to which some subject is considered to be measurable: so that not only length, width, and depth are dimensions of body, but also gravity is a dimension according to which subjects are weighed, speed is a dimension of motion, and infinite others of the same kind. For division itself into several equal parts, whether it is real or only intellectual, is properly a dimension according to which we number a thing; and that mode that makes the number, is properly said to be a species of dimension, although there is some diversity in the signification of the name. For if we consider the parts in an order toward the whole, then we are said to number; if contrariwise we view the whole as distributed into parts, we are measuring it: e.g., we measure centuries by years, days, hours, and moments; if however we number the moments, hours, days, and years, we will finally fill up centuries.
From these things it appears that there can be infinitely many different dimensions in the same subject, and that these add on nothing at all to the things measured out [rebus dimensis], but are understood in the same way whether they have a real foundation in the subjects themselves or have been excogitated from a willed judgment [ex arbitrio] of our mind. For the gravity of body, or the speed of motion, or the division of a century into years and days is something real; not, however, the division of days into hours and moments, etc. All these things nevertheless relate to one another in the same

way if they are considered only under the reason of dimension, as is to be done here and in the mathematical disciplines; for it pertains more to the physicists to examine whether the foundation of these is real. (AT X 447-448)

Descartes points out that although length, width, and depth have a real basis in corporeal things, the three are arbitrarily distinguished.

Here we do not consider these any more than infinite others that are either feigned by intellect or have a foundation in things: as in the triangle, if we want to measure it perfectly, three items on the part of the thing are to be known, namely, either three sides, or two sides and one angle, or two angles and the area, etc.; similarly in the trapezium five, six in the tetrahedron, etc.; all of which can be called dimensions. So, however, that we choose here those by which our imagination will be maximally helped, we will never attend to more than one or two simultaneously depicted in our phantasia, even if we understand that in the proposed thing about which we are busying ourselves there exist as many others as we could wish; for [the essence] of the art is to distinguish these into as many as possible so that we turn simultaneously only to as few as possible but nevertheless to all in succession. (AT X 449)

The notion of dimension established here, Descartes argues, is not the geometer's arbitrary one of the three spatial dimensions length, width, and depth. These correspond to something real in things, he agrees, but they are not simply given by nature; rather, they involve an identification by intellect of a mode or aspect of a thing that is measurable according to some conceivable division. To take the measure of a triangle we need to know at least three elements "on the part of the thing"—three sides, or two sides and one angle, or two angles and the area, and so on. In a trapezium we need five elements (even though it has four sides), in a tetrahedron (a solid with four faces) we need six: "which all can be called dimensions" (AT X 449). Although both the trapezium and the triangle are plane figures existing in two spatial dimensions, the perfect determination requires five and three question-dimensions, respectively; the three-dimensional tetrahedron requires six dimensions.^[24] As is typical of the Regulae, it is not a question of determining or defining the essences of these things. Indeed, their essences in the Scholastic sense are taken for granted, and one's in-

[24] A trapezium is a quadrilateral with no parallel sides; one could determine it precisely by (to give one example) specifying the length of a first side (this is the first factor or dimension), the angles at which the two sides come off the ends of that first (two more dimensions), and the lengths of those two sides (another two, making five in all). A tetrahedron is a solid with four triangular faces; one could determine it by specifying first one of the triangles as a base (this requires three dimensions) and then the apex at which the other three faces meet (by taking one of the sides of the base triangle as an axis and giving three coordinates that would take you from one of its end points to the position of the apex point; this gives three more dimensions).

terest instead is how to construct them unambiguously—more generally, how to get dimensions that codetermine one another and thus allow the solution of the question at hand.

What one needs to solve a question depends on what the question asks for. Moreover, Descartes as always urges us to adapt ourselves to the capabilities of our ingenia. To repeat the admonition of Rule 14:

So, however, that we choose here those [dimensions] by which our imagination will be maximally helped, we will never attend to more than one or two simultaneously depicted in our phantasia, even if we understand that in the proposed thing about which we are busying ourselves there exist as many others as we could wish; for [the essence] of the art is to distinguish these into as many as possible so that we turn simultaneously only to as few as possible but nevertheless to all in succession. (AT X 449)

To summarize the process of Rules 13 and 14: In setting up the problem one uses intellect to enumerate and distinguish all the relevant aspects, terms, and divisions of the thing in question. Then one expresses in easily imaginable, figurate form the proportional relations that are determinately given, and one treats what is unknown as though it were known—known, that is, in relation to the proportional participation in natures of other things or their aspects.^[25] The figures—chiefly line segments, but also discrete magnitudes and two-dimensional figures where appropriate—will then become the objects of manipulation as one tries to equalize the various proportions, with the aim of expressing the designated unknowns in terms of what is known. To keep this all manageable for an ingenium that has difficulty attending to more than two things at a time, one will use marks and symbols to keep track of the problem—a technique that will generate symbolic equations.

But at this point Descartes is envisioning more than the development of modern algebraic problem solving. What counts most for him is the universal relevance of proportional participation in natures, a participation that is expressible in its most perspicuous form not by algebra, which is symbols without content, but by figures, which visibly embody actual proportions. For every algebraic manipulation one can perform on an equation there is, more important, a quite precise, imaginably executable geometric construction corresponding to it. (In fact, it is only in the later mathematics of the Geometry that there emerges an independence, or rather

[25] For example, in a physics problem concerning the weight, density, hardness, and size of minerals, there will be proportional relations between the different aspects (e.g., weight might be equal to length times width times height times density, and hardness might be a function of density). These proportions "reduce" each nature to others, not in the sense of ontologically eliminating it but in the sense of expressing interconnections. The science of the Regulae is not the reductive science of the Discourse.

quasi-independence, of algebraic manipulation from geometric representation; nevertheless, the Geometry is in principle still about how to discover and execute exact geometric constructions by following the pattern of manipulations indicated in algebraic equations.) The marks and symbols are especially helpful in the sorting out of the elements of a problem, for representing in a contentless but mnemonically useful way the line segments and other figures that embody the proportions. The sorting-out process is what the ancient geometers called analysis: taking the unknowns as known and schematically working out what would be true if they were known; and then, provided one reaches the point of having reduced the problem to terms that are known or constructible, the reverse process of synthesis, of the actual construction (or deduction, a leading down ) of the solution, can be carried out.

In comparison to the ancients, Descartes simplifies the analytic phase by making it more accessible to human ingenium. The ancient geometers had worked with complicated figures requiring a sophisticated geometrical insight, and each stage of analysis required finding an executable next step that was usually dependent on multiple considerations (e.g., one might need to add several new lines to an already complicated figure). Descartes uses the identification of relevant dimensions, the simple figurate representation of proportions one at a time, and the symbols and marks of "algebraic equations" to reduce the analysis to a progressive linear sequence requiring attention to no more than two things at once. What is more, the algebraic equations, by containing direct symbolic references to the original givens and unknowns of the problem (x ⁴ = a² + 2bc - c³ directly involves the x, the a, the b, and the c ), allow one, if the analysis is successful, to construct the unknown proportion step-by-step (multiply the line a by itself, add to this result the line that is twice the product of b and c ; subtract line c multiplied by itself and by itself again; then take the fourth root of this result, and you get the desired, previously unknown x ). This entire process, then, is simply an iterated and reiterated application of intuitus and deductio to the image forms of the problem, arrayed in a sequence and manner that makes everything as clear and as imaginable as possible.

The decisive advantage of this kind of deductio over the deduction from axioms and postulates of ordinary logic and dialectic is its perfect reversibility and reciprocity: if y is the product of f and g, it is also true that f can be obtained from y and g by division (i.e., by a proportion, y/g ), and likewise g can be obtained as y/f The terms of dialectic, however, are not in general reversible. "All human beings are animals" does not justify the conclusion that all animals are human beings but only the weak and relatively indeterminate "Some animals are human beings." From this latter assertion no further conclusion is possible without the addition of a new

universal premise telling us something about either animals or human beings. Descartes's problem solving, by contrast, uses universal mathesis to reason about nothing other than particularly represented relations, so that the presence of nothing but particulars does not prevent a solution. On the contrary, it is what makes a single determinate solution possible.

In the concluding remarks to Rule 14, Descartes notes that in choosing among figures for expressing the relations in problems "it is necessary to know that all relations that there can be between entities of the same genus are to be referred to two headings: namely, to order or to measure" (AT X 451). As to order, the whole Regulae has been directed toward showing the nature and consequences of order and the search for it. Originally finding out an order can take a great deal of work (though some orderliness is intrinsically simple), but once an order is discovered there is no difficulty in knowing it by following Rule 7's advice of running through a sequence of the parts that have been distinctly separated from one another,

because, to wit, in this genus of relations one part is referred to others out of themselves alone, but not by means of a third, as happens in measurings, about the evolving of which we are alone here treating. For I recognize what is the order between A and B without considering anything beyond each extreme; I do not, however, know what the proportion of magnitude is between two and three unless it is considered by some third thing, namely, by the unity which is the common measure of each. (AT X 451)

Continuous magnitude can always be reduced, at least in part, to a multitude (a line of length 7.333 . . . can be approximately represented by 7 points or, if one takes. 1 as the unit, 73, etc.), which means that problems of measure can, by means of the unit of measure, be translated into problems of order.^[26] To represent dimensions by continuous magnitude we should not use anything more than length and width; that is, we should not attend to more than two dimensions simultaneously. In conclusion, he points out a consequence: that even problems concerning mathematical entities (like geometrical figures) are to be treated according to this method.

Here propositions are to be no less abstracted from those figures that geometers treat, if the question is about them, than from any other matter whatever; and for this purpose none are to be retained beyond rectilinear and rectangular surfaces, or right lines, which also we are calling figures,

[26] I do not see that one in fact gains much fundamentally from this procedure, since it does not eliminate the unit (which is a third thing interposing itself between the two things to be compared) but rather leaves it implicit. Nevertheless, it perhaps gives a more imaginable precision: two lines, one of length 71/3, the other of length pi, leave considerable work to the imagination in fixing the exact proportion, whereas putting 73 tenth-units next to 31 makes the proportion somewhat more determinately, though approximately, evident.

since by means of these we imagine a truly extended subject no less than by means of surface, as was said above; and finally by means of the same figures both continuous magnitudes and also multitude or number are to be exhibited; and nothing simpler for the exposition of all differences of relation can be invented by human endeavor. (AT X 452)

Thus, far from applying ordinary mathematics to all things, Descartes is advising the application of mathesis universalis, the science of concretely imaged proportional relations, even to arithmetic and geometry. A mathematician will therefore identify relevant aspects or natures in any strictly mathematical problem, proportionalize the givens, then perform an imaginative process of continued proportionalization to solve for an unknown. If this process is different from the ordinary methods of mathematical proof, so be it.

The rules following—15 through 18 have both headings and explications, 19 through 21 only headings—are virtually contained in Rule 14. Rule 15 shows how to use a point or line or square to stand for the unit and then how to extend these to represent measures that are commensurable with the unit or to construct a rectangle to exhibit to the eyes two measures in relation to one another. Rule 16 advises the use of concise marks to keep track of the problem as a whole; Rule 17 advises surveying all the parts of the problem on the same basis of instantiated proportions, whether they be known or unknown. Rule 18 shows how to add and subtract line segments from one another and explains the principle underlying multiplication and division using lines and rectangles. Crucial to the effectiveness of the procedure is the ability to translate a measure from a line segment into a rectangle and from a rectangle into a segment, so that it is never necessary to exceed two spatial dimensions (in ancient mathematics, the multiplication of two numbers was conceived by constructing a rectangle with sides of those lengths; multiplication by a third number required the construction of a solid; multiplication by a fourth number thus presented nearly insuperable problems of conception, since it would have required entering a fourth spatial dimension).^[27] Rule 19 (heading only) says that in solving a problem we need to find as many magnitudes (line lengths, rectangles, numbers) as there are unknown terms by using operations on the unknowns as if they were known (in modern algebraic terms, we look for equations setting x, y, z, and any other unknowns equal to some formula that contains only known values). Rule 20 (heading only) promises to tell how to carry out the operations of multiplication and division

[27] Descartes's technique of reducing a plane figure to a line represents an advance over the mathematical adaptation of the memory art he had explained to Beeckman in October 1628; see chap. 3, Sec. E above.

(which had been put off until this point because of their greater technical difficulty). And Rule 21, the last extant heading, urges that many equations be reduced to a single one. At this point the Regulae breaks off.

F. THE SCIENCE OF ORDER AND MEASURE AND THE PARADOXES OF SIMPLE NATURES

Was the Regulae abandoned, or did Descartes simply leave it incomplete? Are there tensions and conflicts between it and the later philosophy, or is the later philosophy simply a continuation of what was laid down in its rules?

With each question cases can be made for either alternative. Consider the possible significance of the fact that the break in the Regulae occurs at Rule 21 and that none of Rules 19 through 21 consists of more than a heading. All three have to do with forming equations on the basis of figurate and symbolic representations and the operations of addition, subtraction, multiplication, and division discussed in the preceding rules. The last substantive remark made in the commentary under Rule 18 says that in order to reduce any rectangle to a line segment one must be able to construct on a given line a rectangle equal (in area) to a given rectangle. The rationale is this: The product of a length a multiplied by a length b can be represented by the area of a rectangle with sides of length a and b . If one wants to multiply this product by another number, one needs to convert ab into a line length and then construct a new rectangle with sides of length ab and c , the area of which will be abc . One could construct a cube with dimensions a, b, and c, but then in order to perform a new multiplication one would have to reduce the cube's volume to an area or a line; but a cube in any case requires attention to three measures at once (length, width, and height), contrary to the advice of the Regulae to keep comparisons to two items at a time. Thus any geometric technique of multiplying is going to require the reduction of a higher-dimension figure to a figure of the next lower dimension.

If a and b are integers or rational numbers, the construction of a line segment equal in length to their product is relatively simple—one can just reckon up the total number of units or fractional parts and then build up the product line piece by piece—but if one of the numbers is irrational, this will not be possible. Nor will any arithmetic or algebraic techniques (algorithms) give the result exactly, for they work by approximation. The desired line length could be obtained by identifying a general technique of geometric construction, however. Given a rectangle of area ab, if one can construct a second rectangle of equal area on a line segment having unit length, the two sides adjacent to this unit segment obviously will both

have length ab. So the ability to construct on a given unit line a rectangle equal in area to another given rectangle amounts to being able to construct a line whose length represents the product of two numbers.

The general technique is conceptually simple; actually constructing such a line or rectangle is quite another matter.^[28] But does this Regulae method really serve Descartes's larger and ultimate goals? The Geometry of 1637, one of the three essays that followed the Discourse on the Method, gives some insight into this question. On the one hand, the Geometry can be conceived as Descartes's further elaboration of the correspondence between line lengths and symbolic algebra, with the algebra subordinate insofar as it is an easily manipulable symbolic form for expressing proportions that are ultimately to be translated into geometric constructions. On the other hand, it can be seen as a rejection of the method of applying straight lines and rectangles in favor of a general technique of solving for proportions by the generation of complex curves. The Geometry works as resolutely in two dimensions as does the Regulae, but its favored tools are straight lines and curves derived from the motions of straight lines and lower-dimensioned curves by continuous and rigorously mechanical processes. The antecedent of this aspect of the Geometry is Descartes's study around 1620 of the properties of new kinds of tools for geometric construction, like the proportional compass, the operation of which could be used to trace out curves by virtue of the mechanical linkage of its segments. The later geometry of Descartes puts a greater premium on the generative process and its continuity, and it systematically explores the possibilities of generating complex curves that correspond to algebraic equations. This is a dynamic geometry, in contrast to the more static and episodic character of the mathematics in the extant Regulae, where one always stops to compare just one line to another, then performs one or more simple manipulations, then compares the result to one of the original lines or yet another, and so on. The Geometry 's geometry proceeds continuously and dynamically, not step-by-step.

The context of the mathematics of the Geometry is the scientific understanding of extension and matter that Descartes developed for Le Monde, a work begun about 1630, after the abandonment or presumed abandonment of the Regulae. How that work is predicated on a more continuous and dynamic, but also more restricted, power of imagination will be revealed in the next chapter. Before we can proceed to that, we must con-

[28] If you know exactly how long the unit length should be, relative to the lengths a and b, it is easy to construct the line of length ab. Form a first fight triangle with legs of length 1 and a, then draw a line of length b parallel to the side of length i and construct a second triangle on length b similar to the first triangle by drawing parallels to the other two sides. The other leg of the second triangle (parallel to the side a of the first triangle) will have length ab. When a and b are rational numbers it is always possible to construct a line of unit length, but when they are both irrational the whole trick is to construct such a unit length exactly.

clude the discussion of the Regulae by raising some of the major unresolved problems to which it led. They will be presented under two headings: the paradox of the natures doctrine in the context of science, especially in matters of physiology and physics, and the interaction of intellect and imagination. Imagination, it turns out, figures centrally in both.

Rule 6 introduced the natures in the phrase "pure and simple natures," but what they amount to in the first instance are aspects common to many things (à la Rule 14) that are participated to varying degrees. This differential participation gives rise to an order from the least to the most; the things that most participate in—or, better, wholly contain—the nature in question are called 'absolute'. But even in the course of Rule 6 there begins to appear the thornier question of the participation of natures in one another. Rule 8 and, even more, Rule 12 go on to address not just the powers of mind but also the objects of those powers, the natures. The natures are divided in Rule 8 into the spiritual, the corporeal, and the spiritual/corporeal; in Rule 12, into the intellectual, the material, and the common. Both rules distinguish simple natures from complex ones; the latter are either originally experienced as such, that is, as complex, or are composed by the intellect. Falsity exists only in the composite natures put together by intellect.

The second part of Rule 12 in fact offers a presuppositional account of natures, in much the same way that the first part presents an account of the psychophysiology of the internal senses that, like the hypotheses of the astronomers, is intended to make things clearer without a final assurance of truth. The presuppositional account goes as follows. First, the way that natures exist in reality is said to probably differ from how they appear in knowledge, since, for example, real bodies, which are single and simple as things, are understood as somehow put together out of diverse natures. Second, there are three classes of nature, intellectual, material, and common, the last including the common notions of mathematics and reason (e.g., if two things are equal to a third thing, they are equal to one another). Third, any simple nature is evident, known in itself, so that having the slightest grasp of it is to have it complete. Fourth, the simple natures are conjoined either necessarily or contingently; in the first case separating them makes it impossible to conceive the components distinctly. Fifth, the only things we can understand are these simple natures and a certain mixture or composition of them. Sixth, the composite natures are either known as such through experience or are made up by intellect. Seventh, the composites we make come about through impulse, through conjecture, or through deduction (AT X 418-425).

Marion has argued at length that the simple natures of Descartes are inscribed against a background of Aristotelian ontology and epistemology and that in the doctrine of them is contained what he calls the "gray

ontology" of the Regulae, that is, an inexplicit ontological framework that subsequently evolved into the philosophy of Descartes's maturity.^[29] Marion is certainly right to put such weight on the natures for Descartes's future thought. Descartes sought from them ontological support for the method of the Regulae, and the partially developed theory of natures offered there raises themes and suggests positions of the later philosophy without set-fling them.

In the Regulae, Descartes appears to be quite content with the possible discrepancy between natures in existent things and natures in our knowledge that the first of the seven presuppositions allows. Even on the epis-temological level he displays a remarkable casualness about the natures. Although they are to provide the foundations for all knowing, Descartes does not insist on a perfect reductio to them. He is perfectly happy to get on with the task of problem solving by means of pragmatic and heuristic expedients, the most basic of which is the discovery of series with absolutes that are less than absolutely absolute.

In fact this casualness is not unique to the Regulae but is a recurrent trait in all Descartes's writings, a casualness grounded in a conviction, first, that human nature—human ingenium—is constituted so that it can know and learn and, second, that the foundations of knowing are not infinitely or even greatly remote from our capabilities but only a short distance beneath the surface (often buried under the debris of philosophical maleducation). To use an image of the Discourse, most people build on sand, and this is especially unfortunate because one need not dig deep to hit clay or rock. Or to change to a visual metaphor: one can reach the truth relatively easily by looking carefully and clearing away what is extraneous; once one has done this it is easy to see things, and once one sees them one has them. One might recall in addition the recurrent motif of Descartes's writings that says one must once in one's life undertake an examination of one's mind and opinions to find the truth and combine this with the fact that he acknowledges having recognized this need early in his own life yet postponed the examination for nearly a decade (AT VI 10 16-17, 22; VII 17). That it needs to be done but once in one's life indicates that, unlike ancient theoria and medieval contemplatio, the ultimate truths do not open up a way of life that is a continuous gazing on truth, but that once truths are seen they can be used to guide all the concerns and solve all the problems that life presents. Real truth is secure and may be relied on; given God's veracity, it does not even have to be recalled in its full and present evidence but only remembered as having in the past been known. Yet the atheist can recognize the truth just as well as the believer in the moment of evidence; this confirms that truth's foundations are in essential har-

[29] See Marion, Sur L'Ontologie grise de Descartes, esp. 131-148, 185-190.

money with the careful but nonfoundational use of one's knowing powers. Therefore one can know, live, and act in the light of the truth even when one has not looked to the source of the illumination, and one does not need in every case of knowing to look deep beneath the surface of things.

The Regulae appears to be written in this double confidence, that the foundations are shallow but still secure and that the good use of ingenium does not in most cases require going down to the foundations. The hypothetical mode of all of Rule 12, combined with the relative simplicity of the physical and physiological language and themes posited there (and elsewhere, for example Rule 8, which discusses the anaclastic line of optics in physical terms that are more Scholastic than Cartesian),^[30] tends to confirm that the Regulae precedes the deeper investigations of physics, metaphysics, and foundational epistemology that we regard as genuinely Cartesian. But this would be perfectly consistent with an essential continuity of this work with the ontological epistemology of resemblance, which is more coherentist than foundationalist and expresses a confidence that higher truths are adumbrated even in the least things. A Descartes working at the Regulae with this confidence would see the fundamental cognitive task as one of progressive clarification.

Yet it is doubtless right that the question of what in the last analysis we see when we see and what we know when we know induced Descartes to put a term or end point to seeing and to call them natures. His first impulse, one expressed in the main project of the Regulae, was to treat them as the aspects of things; his second impulse, which appears in the obviously incomplete sections of Rules 8 and 12, was to conceive them as a kind of object (or res, thing) capable of participating in other natures just as ordinary things do. At the limit of this participative process stood the ultimate simple natures, which need no further phenomenological or causal support. Descartes's treatment of them is hesitant, however; he does not push the simple natures to their logical and ontological extremes, but instead takes them in several directions not entirely consistent with one another.

The mathetical approach of Rules 13 through 21 is oriented to practical problem solving and an imaginative mathematics of participation in

[30] See AT X 394. The anaclastic line is the curve of a lens that would focus all parallel rays of light to a single point. Although the Regulae 's discussion of optical matters suggests that Descartes had already achieved no little mathematical sophistication in optical theory, his physical theory of light is still markedly traditional. A sign of this is his use of diaphanum. CSM is certainly wrong in rendering totum diaphanum as 'the entire transparent body' and explaining it in a note as 'the very fluid 'subtle matter' which Descartes took to be the medium of the transmission of light' (CSM 1:29), for diaphanum is simply the Latin term for the transparent medium that, according to Aristotle, must be actualized by light for the proper sensible, color, to be conveyed to the eye.

natures that gives rise to measuring the degree of participation. It has a pragmatic conception of natures, understood as dimensions, which can be either fictional or real. The idea of reductio presented is not ontological but proportional, which means simply that proportions allow us to express one thing in determinate, reversible relations to others by way of various dimensions. So, for example, the reflection of a light ray can be analyzed as a relation between lines standing for the ray and a line standing for the reflecting surface; the proportions between the lines can be expressed by angles; and the natural forces involved can be put in terms of the angles and lines. The source of error in this process is the intellect, when it overlooks what the concrete presentation of the problem contains. Nevertheless, intellect still has the fundamental problem-solving task of sorting out the different nature-dimensions in the givens of the problem and determining what is relevant—which also means leaving aside the irrelevant, although no criterion for this distinction is given.

Rule 6 presents an object-centered approach oriented by a participation in natures that gives rise to an ordering, serializing activity. It establishes a wobbly foundation for the later mathetical rules, which presuppose the in-genial activity of comparing one thing to another in light of natures that are the starting point of Rule 6. The serializing is chiefly of the things that participate in natures rather than of the natures themselves. The things rather than the natures are in the first instance treated as absolute or respective. The second annotation of the rule begins to broach the question of the participation of natures in one another without clarifying whether this participation is of a different kind.

Rule 12 (and Rule 8) attempts to probe what Rule 6 presupposes: the character of the physical processes and of the psycho-organic apparatus that convey the appearances of things to mind, and also the very natures that are increasingly conceived as the ultimate objects of the psychological activities. But Rule 12 proceeds by presupposition. In the psychophysical part it presumes that the impressions of things are conveyed by physical processes that are wholly reliable and preserve the semblances of things intact (one might surmise that he expects the physicist to confirm this in short order), and it adapts a long-established internal senses doctrine to a modified but still traditional physiology in a manner that shows no particular acquaintance with concrete anatomical studies^[31] He identifies the real source of imagination as the vis cognoscens acting in the organ phan-

[31] Descartes probably did not begin studying anatomy at first hand until sometime in 1629; see Dennis L. Sepper, "Descartes and the Eclipse of Imagination, 1618-1630," Journal of the History of Philosophy 27, no. 3 (July 1989): 379-4o3; and Sepper, "Ingenium, Memory Art, and the Unity of Imaginative Knowing in the Early Descartes," in Essays on the Philosophy and Science of Rtné Descartes, ed. Stephen Voss, 142-161 (New York: Oxford University Press, 1993). See also chap. 3, footnote 32, above.

tasia—as much will-like as intellectlike, since it is an intentional application of the knowing force—and places thinking's center of gravity there; yet he allows a certain degree of autonomy to a pure intellect that he scarcely even begins to articulate.

In the natures part of Rule 12 he countenances a possible discrepancy between how things in the world and things in the mind participate in natures, yet he postulates a foundationalist ontology of epistemological natures that has to be ultimately consistent with the being of real things if it is to be cognitively effective. He says that natures must be composable and distinguishes between necessary and contingent composition, but he does not clarify necessity in any deeper sense than the mind's inability to distinctly separate the necessarily composed natures. In this schematics of natures as necessary or contingent, falling into the three classes of the purely material, the purely intellectual, and the either material or intellectual, he begins to evolve a higher-order vocabulary for talking about natures that suggests there are higher-level or transcendental simple natures, the character of which is left obscure. Finally, the theory of the composition of natures that our intellect performs, according to either impulse, conjecture, or deduction, begins to address what might provide a foundation for the cognitive activity of imagination, although Descartes's presentation is a mixed bag of problematic assertions (AT X 424). (1) He tells us that impulse can lead to either truth or error: it is never a source of error when it comes from on high (although this stands outside the scope of method), rarely when from our free will, almost always when it is due to an indisposition of phantasia (like jaundice). (2) By conjecture we compose what is merely probable; this does not deceive us so long as we do not assert it to be true. (3) Deductio remains as the sole way of composing things so that we can be certain of their truth, precisely by using intuitus in a progression that reveals the necessity of the conjunctions of natures. Descartes appears to have found the themes of his maturity and the germ of his approach to them in this quick sketch, but the compression and problematic coherence suggest that he had not yet thought them through.

G. COMPARATIVE INTELLECT AND CONCRETE IMAGINATION

Intuitus, of course, is the easy and distinct conceptum of a pure and attentive mind. Yet the thing-oriented intuitus of the early rules as well as the mathetical intuitus of the final rules begin to intimate that natures might be the easy and distinct things grasped by the pure and attentive mind. Consonant with the biplanar consciousness of Descartes's philosophy, in-tuitus is the well-prepared seeing of a thing with respect to a nature. Natures are thus in the first instance a modality of the grasping-seeing of a

thing, an indirect object, so to speak. Can natures in turn become the direct objects of intuitus? If they can, they would resemble the ideas of the later Cartesian philosophy.

There are reasons provided in Rule 14 to think that Descartes did not intend such an outcome, that he in fact intended something quite different. The relevant passage is the long discussion of the mutual assistance of intellect and imagination in conceiving the various senses of extensio, 'extension'. He introduces the passage by saying that the mathetical rules are indeed useful in the mathematical sciences, but that

the utility is so great for attaining to higher wisdom that I am not afraid to say that this part of our method was discovered not on account of mathematical problems but more that these things are to be learned almost solely for the sake of cultivating it. And I suppose nothing from these disciplines except perhaps certain things that are known through themselves and obvious to everyone; hut the knowledge of them as it is usually presented by others, even if it is corrupted by no manifest errors, is nevertheless obscured by oblique and badly conceived principles that we will try to emend here and there in the things that follow.
By 'extension' we understand all that has length, width, and depth, not asking whether it is a true body or only space; nor does it appear to require greater explication, for nothing at all is more easily perceived by our imagination. Nevertheless, because often the learned use such acute distinctions that they dissipate the natural light and discover shadows even in those things that are never unknown [even] to peasants: they are to be advised that here by 'extension' is designated not a distinct something, separated from its own subject, nor do we acknowledge in the universe philosophical entities of this kind, which do not truly fall under imagination. For even if someone could persuade himself, e.g., that it would not be contradictory, if whatever is extended in the nature of things were reduced to nothing, that extension itself existed per se alone, he nevertheless will not be using a corporeal idea for this conception but only badly judging intellect. Which he will confess him-serf, if he reflects attentively to that image itself of extension that he will try to feign there in his phantasia: for he will notice that he does not perceive it destitute of every subject, but imagines it completely otherwise than he judges; so that those abstract entities (whatever intellect should believe about the truth of the thing) nonetheless never are formed in phantasia separate from subjects. (AT X 442-443)

It is the intellect, not the imagination, that is responsible for error. The passage discusses extension in particular, but its conclusion applies to all thinking directed toward an image, all thinking that requires or involves phantasms. The intellect, not the imagination, tends to mislead us because it often looks away from the phantasm when the conclusion it tries to draw concerns the phantasm. In its desire to make subtle distinctions the

intellect dupes itself; it overlooks the fact that it is trying to separate the inseparable^[32]

These reflections about what a word is taken to mean versus what even the most unlearned peasant knows about the thing the word designates lead to a discussion of the various senses in which imagination and intellect take things in different contexts of meaning.

Since henceforth we shall be doing nothing without the assistance of imagination, it is worth the effort to distinguish carefully through which ideas the individual significations of words are to be proposed to our intellect. For which purpose we propose to consider these three forms of speaking: "extension occupies place," "body has extension," and "extension is not body."
Among which things the first shows in what way 'extension' is taken for that which is extended; for I plainly conceive the same thing by saying: "extension occupies place," as by saying: "the extended occupies place." Nevertheless, it is not on that account the case that to avoid ambiguity it is better to use the word '[the] extended': for it would not signify so distinctly that which we conceive, namely, that some subject occupies a place because it is extended; and someone could interpret [it] as "the extended is a subject occupying place," no differently than if I should say: "the animated occupies place." Which reasoning explicates why here we shall say that we are dealing with extension rather than with the extended, even if we think it should be conceived no differently than the extended.
Now we come to these words: "body has extension," where 'extension' we understand to signify something other than 'body'; we nevertheless do not form two distinct ideas in our phantasia, one of body, the other of extension, but only a single one of extended body; and it is no different on the part of the thing than if I should say: "body is [the] extended"; or rather: "the extended is extended." Which thing is peculiar to those entities that are [= exist] only in another, and cannot ever be conceived without a subject; and it happens otherwise in those things that are really distinguished by means of subjects^[33] for if I should say, e.g.: "Peter has riches," the idea of Peter is plainly different from that of riches; likewise, if I should say: "Paul is riches," I imagine something completely different than if I should say, "riches are riches." Many, not distinguishing this difference, falsely opine that extension contains something distinct from what is extended, just as the riches of Paul are other than Paul.
Finally, if it is said: "extension is not body," then the word 'extension' is

[32] Abstractions and distinctions of reason therefore can easily turn out to be false when they concern concrete objects or images. The implications of this could be considerable, especially in the search for truly fundamental simple natures.

[33] CSM translates "illis, quae a subjectis realiter distinguuntur" as "entities which are really distinguishable from their subjects" (CSM 1:60), hut the examples are not of things that are distinguishable from their subjects (Peter, Paul, and riches) but of two subjects that are distinguishable.

taken far differently than above; and in this signification no peculiar [= particular] idea in phantasia corresponds to it, but this entire statement is accomplished by pure intellect, which alone has the faculty of separating this kind of abstract entity. Which thing is the occasion of error to many, who, not noticing that extension taken in this way cannot be comprehended by imagination, represent it to themselves through a true idea [veram ideam]; since such an idea necessarily involves the conception of body, if they should say extension so conceived is not body, they are imprudently implicated in this, that "the same thing is simultaneously body and not body." And it is of great moment to distinguish statements in which names of this kind: extension, figure, number, surface, line, point, unity, etc., have so strict a signification that they exclude something from which they are truly not distinct, as when it is said: "extension, or figure, is not body"; "number is not numbered thing"; "surface is the limit of body, line of surface, point of line"; "unity is not quantity," etc. All which similar propositions are to be completely removed from the imagination in order to be true; for which reason we shall not be dealing with them in the following.
And it is to be diligently noted that in all other propositions in which these words, although they retain the same signification and are said in the same mode abstracted from subjects, nevertheless exclude or negate nothing from which they are not really distinguished, we can and must use the help of imagination: because then, even if the intellect precisely attends to only that which is designated by the word, imagination nevertheless must feign a true idea of the thing so that, whenever use demands, the same intellect can turn to its other conditions not expressed by the word and not ever imprudently judge that they have been excluded. So if the question is about number, we shall imagine some subject measurable by means of many units, to the sole multitude of which the intellect can indeed turn in the present moment, yet we shall take care that from this subsequently it not conclude something, in which the numbered thing is supposed to have been excluded from our power of conception: as those do who attribute to numbers wonderful mysteries and unadulterated nonsense, to which they certainly would not lend so much faith if they did not conceive number to be distinct from the numbered things. Likewise, if we are dealing with figure, we will think we are dealing with an extended subject according to what is conceived by this reason only [sub hac tantum ratione concepto], that it is figured; if with body, we will think we are dealing with the same thing, as so long, so wide, and so deep; if with surface, we conceive the same as long and wide, depth being disregarded, not negated; if with line, as long only; if with a point, the same with everything else omitted except that it is an entity. (AT X 443-446)

This extended passage is motivated in the first place by the need to explain how terms in linguistic propositions are to be conceived by the mind, in particular in what ways the imagination can conceive them. As corollary, it also indicates a role of pure intellect, the functioning of which Descartes had in Rule 12 promised to describe. The particular context in which the passage appears is the preceding statement that all the proportions that

can be detected ought to be expressed in extension and figuration, so that it appears to be the appropriate place for an explanation of 'extension'; yet the points made are not restricted to this notion alone but are applicable by analogy to distinctions of other terms as well.

The passage, in brief, says the following: Every word or form of signification has a "true (imaginative) idea" associated with it, and in almost all cases it is appropriate and necessary to portray this idea in phantasia. A statement that has two terms (e.g., "body is extended") does not necessarily require that two things be imagined, however, because certain natures are such that they are always and everywhere implicated with others. In such cases we should form one image or idea, and by a difference in focus, by a different concentration or grasp—that is, by a different concept —the intellect can now direct itself to this or that aspect of the one image. In other cases, when there is not this implication of natures in the two terms (e.g., "Peter has riches"), it is necessary to form two ideas. But there is a class of cases where meaning or signification is intended in so restricted a sense that it is inappropriate to form an image because the statement would be contradictory if it directly intended the image. In these cases it is pure intellect alone that makes the judgment truthfully. To put this latter point differently, when we say that extension is not body, we are referring precisely not to either extension or body but to the distinction between the two, which (because both the subject and the predicate actually involve extension) no image can directly present and which only the intellect can grasp securely.^[34]

For the second time in the Regulae the role, or at least one of the functions, of the pure intellect is explained (the first was in Rule 13, where pure intellect was assigned the task of setting up the initial terms of a problem). It might appear on first glance that it demonstrates the power of the intellect to rise above imagination. But the tenor of the passage is at odds with this conclusion. It rules out, in general, philosophical entities that do not fall under the imagination, that is, abstract entities that are entirely separated from a subject, from something in which they inhere. When there is something imaginable or portrayable in imagination, the intellect does wrong, and risks error, by trying to judge on its own. In prescinding or turning away from the true image the intellect tends to overlook what is actually included in the idea; it is as though the intellect can easily "forget" when it is not presented with an image, or with some other kind of concrete object. Moreover, the kind of case in which the passage allows

[34] Descartes does not so much forbid one to imagine anything in this case as consider the imagined things an indirect object that must be looked away from in order for the intended negation to be true; or, more precisely, it is to its own way of taking and not taking images that the intellect must directly attend, not to the images themselves.

knowing by pure intellect is a very limited, negative knowledge. All the examples given involve privation or negation (note that "surface is a limit of body" and the like involve the privative notion of limit). Some insight into the issue can be gained from the statement that, in the case of "extension is not body," pure intellect alone "has the faculty of separating this kind of abstract entities." It is only in such an abstract realm, in the abstract, indirect taking of things, that these statements are true.

Behind this assertion is probably the thesis of Aristotelian-Scholastic philosophy that a faculty cannot judge of forms in a way that exceeds its proper powers. Vision can perceive white, it can even distinguish white from black, blue, red, and so forth, but it cannot judge that a white thing is sweet; for this there is required the common sense. Common sense and the other internal senses can judge of such particular differences but not of the genera or forms; for this one needs intellect. Yet intellect does not operate without a phantasm; that is, thought is intellect busying itself about a phantasm in order to grasp what that phantasm stands for by means of the form actively abstracted in intellect.^[35]

Rule 14 agrees insofar as it is the intellect that is properly said to understand by attending to the true ideas in phantasia. One is able in phantasia to make separate images when the proposition in question allows it; when the proposition is about different aspects of an inextricable whole, then the differential attention of intellect alone can track the difference. But when intellect goes it alone, it is not capable of making positive judgments truthfully; rather, it is only able to note differences by way of severely restricting the meanings of terms and abstracting them from all images.^[36] This would seem to be connected with the intellect's capacity of noting simple natures: when properly prepared it can recognize them in things, and it can recognize the difference between such natures, but it cannot define their essences. Properly prepared intellect can only experience simple natures; there is no deeper sense in which they can be said to be "known" by intellect. But this is to reinforce that intellect's function is to enable us to distinguish and recognize, to grasp similarities and differences. In comparison to some other traditional notions of intellect this

[35] In this respect, Descartes arrives at a similar conclusion, in that he denies that one can ever come into possession of a nature or an abstraction without the direct—or, in the case of negations, indirect—presence of a phantasm.

[36] This helps throw light on the point Descartes made in Rule 12 (AT X 418—419) about limit not being a simpler nature than the things from which it is abstracted. Although he was willing to allow that the negations of simple natures are themselves simple natures for methodological convenience, limit is an act of negation or privation that can be applied indifferently to this or that nature. Limitation is a fundamental act of intellect rather than a simple nature that appears in itself.

seems not very powerful, but it nevertheless is, precisely because it allows a freedom from the brute force of the given taken as an undifferentiated whole, or as a whole differentiated only by corporeal processes (as in functions of the brain and nerves). Whatever is presented to us we can compare or contrast with other things not immediately present, things we can call up from memory or produce by the recombinative powers of imagination—that is, of the knowing force acting in phantasia. Intellect allows us to articulate thoughts directed to things, but when left to itself it is mute and inarticulate. In saying that "extension is not body" intellect is in effect judging that its way of attending to things is differentiated; that is, this judgment is not about the thing but about the intellect's way of acting.

This leaves a problem, however: can all these things be said of the intellect when it recognizes what is entirely noncorporeal, for instance, what willing or ignorance is, or that my being implies God's? The answer, I think, is twofold. One part has to do with our expectations of Descartes. Insofar as we judge from our knowledge of what he wrote later, we want to find anticipations of it earlier. There is nothing wrong with this; indeed it is probably a necessary prejudgment that induces us to read the noncanoni-cai works of any major author. But we must avoid the pitfalls of an excessively teleological reading, that is, assuming that the later positions are present in nuce in the earlier or that they were an inevitable outcome. Although the Descartes of the Regulae has set out on the way of ideas, it is not by the way of doubting, a via negativa, but by the via positiva of knowing. Although he understands intellect to be as different from body as blood is from bone (AT X 415), this is not yet the basis of a truly dualistic metaphysics. Although the intellect is what knows in the proper sense, the activity of pure intellect that he describes in Rules 13 and 14 is only a momentary step back from the imaginative realm, and nowhere does he state or even suggest that dealing with problems involving perfectly intellectual or spiritual natures is substantially different from dealing with other natures, especially since there are similarities and analogies that he is willing to countenance between the spirit and the body.

The second part of the twofold answer about the incorporeal has to do with what we have learned about intellect's acting on its own. It acts proximately and for the most part in conjunction with phantasia. It can abstract from phantasia, but then it risks error; what it can truthfully achieve in prescinding from the corporeal is an awareness of its own way of attending to and grasping the corporeal. When it truthfully judges that "extension is not body," it is not attending to the corporeal ideas but rather to its way of attending to ideas, and in this sense such a judgment is still related to the corporeal; it is a potential involvement with the corporeal rather than an actual involvement. For example, intellect would not truthfully be judging

that extension is not body if it made such a judgment having never experienced the extended realm or not recalling what kind of experience the judgment is about.

But then it is clear that intellect can attend to its own activity with images (what Descartes refers to as its reflex contemplation of itself; AT X 422-423): its knowing, not knowing, doubting, willing, to name the four that Descartes explicitly mentions (AT X 419). Most of this attention to self will be related to, if not about, activity in the body (which includes sensation, imagination, and locomotion). The question then becomes: is it possible for intellect to attend to itself in complete abstraction from the body and imagination? For the Meditations, the answer would seem to be a decisive yes,^[37] but the answer for the Regulae is less clear-cut. It is evident that there is a "depth dimension" to intellect attending to itself, for the case of Socrates, who by doubting that he knows anything comes to know that he doubts, implies that there is an intrication of natures in the purely intellectual realm (AT X 421). Even more suggestive are Descartes's assertions that I can know that because I exist, God exists and that because I understand, I have a mind distinct from body (AT X 421-422). They lead us to suspect that Descartes had already worked out the arguments of the Meditations. Yet important elements are missing. The argument for God's existence requires a distinction between the formal and objective reality of ideas, of which there is no trace in the Regulae , and the Meditations' radical distinction of thinking and extension is present in the Regulae in at most attentuated form. Since God as the foundation of our personal being is a very old theme in Christian thought—not to mention the contrast of body and soul—with a particularly pertinent form of it in the arguments of Augustine,^[38] the notion that our existence implies God's is not in itself distinctive or remarkable. If the method of the Regulae is any indication, Descartes probably thought that such truths were potentially evident by intu-itus. By not confronting the questions of existence the Regulae prescinds from these matters, and it leaves open where, how, and at what level natures are implicated in one another.

Descartes justifies Rule 14's lengthy discussion of intellectual and imaginative approaches to extension by expressing his fear that it is not sufficiently long to prevent people from falling into error, "so preoccupied are the ingenia of mortals" (AT X 446). The practitioners of geometry and arithmetic ordinarily think they are dealing with numbers abstracted from all subjects, lines that have no width, surfaces that have no depth, and so on, all things that he has shown to be false. Rather than go on with such a critique, he turns in the rules that follow to a positive account of how objects should be conceived, that is, imagined.

[37] Chapter 7 will give us some reasons to be less certain of this, however.

[38] For example, in book l0 of the Confessions.

The method of the Regulae is calculated to explicate how we can understand things through the aspects they display. It is a rationalization of the coherentist psychology of resemblance characteristic of his earliest thought. But it is also the beginning of the reification as natures of the aspects under which things appear. What a nature is, is ambiguous: it is an aspect, a way of taking a thing, but it is also ontologically ambiguous and appears to take on an increasingly independent existence. The evolution of the natures doctrine did not have solely ontological consequences, of course, for "natures" is also the answer to the question, "What is (most) knowable?" As the natures underwent a change in conception, so too did the faculties of imagining and knowing.

H. THINKING IMAGINATION BEYOND THE REGULAE

Thinking, as the young Descartes conceived it, was an activity, one that was constantly challenged to respond to the changing deliverances of the sensitive faculties. (Consider, for example, the Compendium musicae's dynamic, imaginative way of conceiving a song out of the individual notes.) In the Regulae this activity was conceived as having two major cognitive forms, intuitus and deductio. Discursive reasoning is described many times as a movement of cogitation, once even as a movement of imagination. Since neither intuitus nor simple deductio can be taught, the rules were designed to call attention to them and how they might be practiced and honed and to regulate those activities of cogitation requiring a complex movement of thought from one thing to another.

Intuitus might seem to be perfectly passive, but in fact it is defined as a grasping-beholding by a pure and attentive mind. The mind and its object must be purified and prepared, attention must be carefully directed, and the mind must take hold of its object. Because of this activity Descartes did not need to make the distinction of the Fourth Meditation between the intellect that perceives ideas (without assent or dissent) and the will that makes judgments. The mind or, more significant because of the implications of its name, the vis cognoscens, knowing force, brings a significant component of will into knowing and implicitly makes it more activity than receptivity.

The vis cognoscens can act on its own, but the greatest part of its activity is in and through phantasia. Accordingly, the model for knowing that the Regulae presents is a thinking through images: the distinct and easy grasp of the aspects they present, and the knowing power wending its way among images—noting, comparing, forming, and transforming them—in order to arrive at the truth of any question.

In the accounts of cogitation and imagination given in the Meditations, there is a difference of no little significance. Although imagination is men-

tioned as part of thinking in the Second Meditation, in the Sixth it is excluded from the essence of the thinking thing; and in the Third Meditation thinking is apparently transformed from the active grasping, comparing, and moving of thoughts into the more passive perception of ideas that are in oneself or one's mind innately. One perceives ideas, one has them. Imagination itself thereby becomes just another way of having ideas. Imagining is simply contemplating the shape or image of a corporeal thing (AT VII 28); when the mind imagines it turns toward its associated human body and looks at something in the body that conforms to an idea understood by the mind or perceived by the senses (AT VII 73). Rule 12 had defined imagination as an activity, as the application of the vis cognoscens to the phantasia in order to form new figures. Where has the activity of thinking and imagining gone?

As we shall see in the following chapters, the activity of thinking (and the activity of imagining) did not disappear in the later Descartes but was obscured by the static implications of thinking and imagining understood as the perceiving of ideas. In depicting these activities the later Descartes's vocabulary displaces a psychology of faculties with a unitary power (the mind) standing opposite and viewing a relatively fixed object. This new psychology nevertheless falsifies the genuine logic of the Meditations, which, as the long philosophical and theological tradition of meditation would lead one to expect, requires a highly active, discursive process of thinking through which one is able to perceive certain unities and finally arrive at a contemplative state.

The vocabulary and psychology of the Regulae are closer to the truth than the Meditations. Yet one must also concede that the psychological logic of the Regulae, with its division of the act of discursive knowing into parts, favored the reduction of knowing to the simple mental having of elemental ideas, and by undermining memory the Regulae left the way open to a static intellectualism devoid of concreteness. This, I contend, was contrary to Descartes's deepest intentions, whether we look to 1628 or 1641.

The having and manipulating of images becomes, in the later Descartes, a source of uncertainty. In important respects the later philosophy deprives imagination of cognitive force. Yet in the later Descartes the very activity of meditating is modeled on imaginative discursiveness, and imagination retains cognitive relevance precisely as a dynamic, productive activity. This occurs first and foremost in his mathematics and physics, beginning with Le Monde . So it is to Le Monde that we turn in order to begin locating the traces of imagination in the later Descartes.