Reaccommodating the Original Theory of Equations By Means of a Prismatic Logic
by Robert H. Schmidt
There is a well-known statement by Descartes - in defence of the charge that he had added nothing to what Viète had already done in algebra - to the effect that he began his own theory of equations where Viète left off at the end of his treatise on the Emendation of Equations. Now, the last propositions in that book assert that the roots of several types of equations can be explicated by each of the homogeneous elements that are assumed to make up the coefficients of those same equations, according to a uniform pattern. The modern theory of equations, initiated by Girard, Descartes, Harriot, DeBeaune and others, would make a converse statement, and claim that the coefficients of an equation so affected could be expressed with such functions of its roots. It then desires to extend this approach to all equations. But note that Viète himself considers these theorems to be anomalous or singular, meaning that it is not in general possible to write analogous theorems for equations that are similar except for a different distribution of signs. His general way of constituting equations, which was in fact the first theory of equations, involves relating them to proportions, wherein some of the terms are not solutions of the equations themselves. Why did Viète's successors prefer a root-oriented theory, and what was at stake when they began to favor it?
Both Girard and De Beaune thought that, in general, Viète made a mistake by first resolving the coefficients of an equation into certain given elements as the first step in relating the roots to the coefficients via proportions. They both thought that the true nature of the equation, its true constitution, would be better understood by directly expressing the coefficients as functions of the roots (factions in Girard's parlance). Later, perhaps they could be related proportionally to given quantities implicit in the coefficients. Hence the conversion of Viète's theorems, as stated above.
But both Girard and De Beaune had misunderstood what Viète meant by the constitution of an equation. They thought that he was seeking the ultimate component elements of the coefficients of the equation, and thus of the equation itself; namely, the constituent elements of the coefficients. In fact, he really sought certain elements that would be pertinent to the equation and yet could be set together - that is, constituted-into some relation of a higher order, as in a proportion. (See glossary under CONSTITUTION.) From their point of view Viète had not found the true constituent (that is, component) elements, perhaps because they saw that he needed to rely partially on elements that were extrinsic to the equation under consideration; that is, terms in the continuous proportion that are not immediately component elements of the coefficients, but are placeholders that help make the power when the proportion is resolved into an equation. (Viète is alone in using the term resolution to describe the movement from proportion to equation. It is a further indication of the change in the meaning of constitution that the movement from equation to proportion later receives the name resolution, for the reason that the coefficients are resolved into their parts. See Schooten's little treatise, De Concinnandis Demonstrationibus Geormetricis ex Calculo Algebraico, for example.) Finding a way to relate the coefficients directly to the roots of the equation itself seemed to them a more intrinsic method of constitution. Nowadays we take this desideratum for granted to such an extent that we are no longer even aware that such a decision once had to be made; we have forgotten that the ultimate elements were once thought of as independent of the equation itself.
This misunderstanding set the theory of equations off in an entirely different direction. Unfortunately, Viète's original concept of constitution got left behind. It is unlikely that it would ever have been rediscovered by any natural progress or extension of the root-oriented theory itself, for the simple reason that in such a theory the component elements (the roots themselves) have no mutual relations to one another beyond their being solutions to the same equation. Thus, this concept of constitution is not really essential to a root-oriented approach, and would probably not have suggested itself. And even if it had, concomitant concepts would have made it too uninteresting to pursue. For example, to the extent that the component elements have relations to one another, they cannot in general all be roots of the equation under consideration, and this would seem like a defective situation to an advocate of a root-oriented view.
It is hard for me to see the rise of the root-oriented view as a straightforward case of mathematical progress. For one thing, it committed us to a course of introducing (or at least acknowledging) new entities for the sake of a neat overall pattern. Tidiness is clearly the main motive for Girard. And thereby a legitimate mathematical problem got left behind in the initial stages of its solution - namely, the task of devising a general theory of equations that would not need to resort to the introduction of new kinds of elements.
We also closed a window that allows another aspect of equations to be seen. Certain distinctions may be seen from there that are simply not visible from the root-oriented point of view. (See discussion of SYSTATICAL VS. EXEGETICAL AMBIGUITY in the glossary for an example.)
But above all, I regret the disappearance of the species concept from algebra. The elaboration of the root-oriented theory completely deprived us of it. The new theory called for a more natural way of getting from the roots of the equation to the equation itself than the method Viète had used. Viète derived the equation by algebraic argument on the unknown of a carefully selected problem, but the problems get a little contrived if all the unknowns in the problem must be solutions of the equation to be derived, as is the case for the root-oriented theory. (See Girard.) The new way generated the equation by the multiplication of factors. Now, few if any of Viète's contemporaries or interpreters had understood what he meant by species. With the link between equation and problem broken, it became all the harder to interpret the letters as species, or as the roles played by the different elements in a problem. Equations could no longer be understood as theorems. Furthermore, factoring involved setting all the terms equal to zero. Accepting zero as a term in equations meant reenforcing the view that the letters in algebra stood for quantities. The original motive for introducing letters into algebra was no longer in force, and new ways of accounting for their significance were found.
Yet there is a fortunate side to this misunderstanding as well. If Viète's ideas had been widely circulated, but not misunderstood, chances are that the root-oriented theory could never have been developed at all. The situation in which the constitutive (or ordered) elements are all roots of the same equation provides the least degree of order among the constituative elements themselves, and could only provide a defective kind of constitution. Any investigation of this situation would have remained within the horizon of the concept of constitution, and having the constitutive elements be as intrinsic to the equation itself as the roots themselves is not a desideratum for a constitutive view like Viète had. The propositions at the end of On the Emendation of Equations may well have remained interesting curiosities. So here again we might have been slaves to one way of looking at equations, unaware even of the existence of alternatives.
It seems that neither one of these theories was actually a direct and logical development of the other. Nor is one of them implicit in the other as a possible generalization or specification of one of its essenttial characteristics, since the corresponding concepts are the presence and the absence of a certain characteristic. And since one of the corresponding concepts is always negatively defined in terms of the other, we could only abstract out a common characteristic by favoring one of the theories, so there is no general theory of which these two theories are mere specifications. We cannot even regard one of the theories as a negation of the other (and thus in some sense still bound to the same conceptual level), since there is one of the negatively defined limit cases in each of the theories.
In short, these two theories have nothing more in common than a common interest in some of the same properties and characteristics of equations. It is hard even to find precise grounds for contrasting them. The problem is that we are not comparing or distinguishing theories about different objects or classes of objects, like two modern style algebras or geometries whose elements are defined by their respective systems of axioms, and which are different or alike to the extent that they have different or like objects. We are not talking about the different hypotheses that could lie at the basis of such a science. Here we have instead two radically distinct points of view for the same object of investigation - namely, the equation itself.
I would also argue that neither one of the above is the perfect theory of equations. We cannot even easily say whether one theory is better than another. The question is rather, under what circumstances COMPONENT elements (as in the root-oriented theory) or CONSTITUTIVE elements (as in Viète's view) are better for understanding a given property of equations. Each opens up a special window on the properties of equations with its own unique vantage point. What from one point of view may appear to be a single tree with a smudge of shadow behind it, from another may be seen as two distinct trees. I would like to find a way of keeping both windows open.
For Viète, Girard, De Beaune and others, a study of the properties of equations had to be preceded by an investigation into the best way of recognizing the equations themselves. This meant finding out what the equation was really about regardless of the particular problem that gave rise to it - in other words, what its nature was. A theory of equations would then display these properties as necessary consequences of these natures.
However, if an equation can in fact show itself in distinct and equally fundamental ways, then any thinking and speaking that traps the attention on one of these aspects is in a certain sense inadequate to its object, and to this extent not true. What we want is a way of thinking and talking about equations that will keep us alive to everything the equation can be. One way of doing this is to reconceptualize the two theories as true alternatives. If this is successful, then, even while the mathematician is thinking the basic principle of one theory, he will be kept aware that there is another principle, that he has favored another one over it, and that it is truly distinct.
I will conceptualize the idea of alternativeness itself with the metaphor of a spectrum. My intention is to reformulate the basic principle of each theory so that the natures favored by the respective theories appear at the extreme opposite ends of a spectrum of possible natures. The spectrum is a good metaphor for what I am trying to do here, because light both reveals other things and yet can be an object of study itself. Also, the different colors of light make the same things appear different. Thus, the logic I am advocating should be described as PRISMATIC rather than focal.
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According to the earlier discussion, the two theories may be distinquished as different interpretations of the 'of' in the expression 'constitution of equations'. In the root-oriented case the equation is the object of the activity, and we think of constituting the equation itself by composing its coefficients out of roots. In Viète's view we think of constitution as something belonging to the equation, setting the elements together into an order that is pertinent to the equation. To this extent we are justified in saying that these two theories are intrinsically different.
However it is not enough to merely distinquish them in this way, saying, on the one hand, that the proper constitutive elements for the equation are elements in a proportion, and on the other, that the proper constituent (component) elements of the equation are the roots. By the very fact that these principles are precisely stated, if we were to keep them as our basic principles, the ultimate effect would be disjunctive - to the extent that we would think of one theory, we would be oblivious to the other. After all, it was because of this that Viète's theory was able to be forgotten.
But suppose we find an unambiguous concept whose compound structure leaves room both for the idea of a constitutive element and the idea of a component element. When the basic principles of each theory are reformulated in terms of this concept, the effect should be conjunctive. That is, thinking the basic principle of one theory would oblige us to also think along lines that are generally characteristic of the other theory, even though we would not see the specific form of that other theory. This procedure does not replace the earlier precision with ambiguity, but rather a narrowness of vision with a kind of peripheral vision. We are made aware that an alternative principle may be discerned if we look intently in a certain direction.
The constitutive elements and the component elements of equations were presented as the candidates for the fundamental nature of equations by the two theories respectively. Constitutive elements went along with the idea of a nature as something that underlies a thing, which is thus in a sense separable and independent of it. Component elements presumed the idea of a nature as having to do with the ultimate parts of something, which are thus intrinsic to that thing. But each theory of equations emphasized one type of nature at the expense of the other. My reformulation of the basic principles as alternatives commences by reenforcing the concept of an element with the concept of a REFERENT, so that the natures need not be thought of disjunctively. Calling something a REFERENT implies that it has a certain independence from the thing being referred to it. In Viète's theory of equations, the elements to which the equation is referred are terms in a continuous proportion. They possess a maximum degree of interrelatedness, hence a high degree of autonomy and definition. When this proportion is proposed as an actual problem about magnitudes, the equation may be derived from it by algebraic or zetetical argumentation, so the proportion is prior to the equation, and independent of it in a superlative manner. In fact, Viète thinks of consitution as invoking a real underlying subject for the property stated in the equation.
But the root-oriented theory does not immediately meet the first requirement of reference, because the coefficients are understood in terms of the very quantities that they determine in the equation, which are thus dependent on them. In order to bring the root-oriented theory into agreement with the requirement of independence, we can reconceptualize the coefficients as functions of separate referent elements that have no specified relations of size to one another, and also understand the unknown X as a function, in the sense that here it must be indifferently equal to anyone of these referents alone. Detaching the referents in this way in no way changes how intrinsic the referent elements are to the coefficcients. The coefficients are now functions of referents, but the root of the equation is equal to anyone of these referents alone. So from the point of view of this reconceptualized theory of equations, X does not immediately signify the sought quantity that satisfies the equation, but is rather a certain function of the referents. The equation is thus a theorem about the referents and is either true or false.
Even though the exact idea of constitution in Viète's sense is not immediately apparent here, because the concept of the interrelatedness of the referents is only present in negation, still it is not made inaccessible, for, once we have introduced independent referent elements, we are eventually led to wonder about what happens when we do specify relations of size among the referents.
However, neither does the constitutive theory as it stands meet the second requirement of reference. An individual letter in Viète's logistic of species is meant to call attention to the role that a certain magnitude has in a certain problem - its functional relationship to other magnitudes in that problem. It performs this service by being a more easily manipulated COUNTERPART subject capable of playing an analogous role. (See glossary under 'SPECIES'.) An expression composed of two letters conjoined in a certain way - we might almost call it a syllable - might be counterpart to a magnitude that functions as the sum of two others, etc. Thus the functional relationships on the side of the counterparts are analogous to the functional relationships on the side of the mathematical beings themselves. If we want, there is no difficulty in calling the single letters the component elements of the syllabic expression.
The problem has to do with Viète's intention in the Recognition of Equations. An equation derives from a particular problem by algebraic argumentation on the unknown, this argumentation having a view to the species of the magnitudes in the problem, or the way they show themselves in the problem. Conversely, recognition is the procedure of invoking prototype group of interrelated magnitudes as the subject of a plausible problem that could lead to a given equation. In the initial statement of his equations, Viète already has their homogeneous terms resolved into functions of individual letters. Recognition then refers these individual letters to the real magnitudes in the prototype proportion, say, as the real subject underlying the property being expressed by the equation. Recognition thus crosses over from the counterpart realm to the realm of real mathematical beings like magnitudes. It has nothing to do with the ultimate parts of the terms in an equation. The constitutive elements of the letters in an equation are more like an underlying subject for those terms. On the side of the counterparts themselves, the letters are considered to be elementary and not themselves functions of component elements in the form of other letters.
However, the nature that is intrinsic to the equation itself should not be found on the side of the mathematical beings themselves to which the letters are counterparts, but rather on the side of the counterparts. But by transferring our attention to the counterparts of the porportional magnitudes themselves, assigning other letters to them if they were not already so assigned, we can think of the individual letters in the equation as functions of these counterparts regarded as ultimate elements. And we do not need to sacrifice any of the interrelatedness characteristic of the proportionals to do this.
With this reconceptualization, intrinsicness in the sense of being a component part is introduced into the constitutive view, and with it an occasion for asking how to make the parts less far removed from the coefficients which are functions of them - a condition for developing the root-oriented view.
So far reconceptualization has succeeded in disentangling the referent elements from the equation in the root-oriented view, and bringing the independent elements of the constitutive view close enough to the equation so that it is possible for them to be called referents at all.
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There is another step we can take to more fully reconceptualize the two theories as alternatives, still guided by the metaphor of a spectrum. This involves reformulating the basic principle in such a way as to emphasize that we are favoring the one type of referent over the other. We can do this with quantifiers like 'all' and 'none', and with comparative and superlative adjectives. When the basic principle is reformulated in such terms, we have first a kind of glance-back-over-the-shoulder to get a glimpse of the other choice, and then a reaffirmation of the present choice. This gives us a new awareness of the alternative. We can tell that it is in a place from which we would have to advance in order to get where we are now. We have a sense of direction. But since there is the same sense of advance when the basic principle of either theory is reformulated in this way, the basic principles are actually oriented in opposite directions. Consequently, the true face of the alternative is always turned from us. In other words, this kind of reformulation has the effect of making us aware of the type of referents used in other theory from the non-characteristic backside of their structural compound.
The basic principle of the root-oriented theory thus becomes 'all the referents are roots' or 'the referents are exclusively roots'. The modern mathematician would not think of adding the word 'all' to his basic principle that 'the component elements are roots', because from his way of viewing even though it is true, it is trival. But once the basic principle has been reconceptualized in terms of referents, the effect of this should be to make us wonder momentarily about the possibility of a theory in which some or none of the referents are roots. This is in fact just about the way Girard formulates his principle in opposition to Viète, although today we have forgotten this.
The basic principle of the constitutive theory becomes 'the referents are ABSOLUTELY ORDERED elements in a continuous proportion' instead of simply 'the referents are continuous proportionals'. The effect of this is to make us wonder about the possibihty of a theory in which the referents are either relatively ordered or not ordered at all. I myself could not avoid placing such an emphasis on Viète's basic principle when I became fascinated with it as an option to the root-oriented approach in which the referents have no order, although I am sure that Viète, lacking the concept of a referent, would have considered the reformulation redundant.
In short, it is hoped that addition of the word 'all', the same word that was used historically to herald a new theorizing about equations, will help to revitalize the sense of choice in our modern root-oriented theory of equations. Similarly, in the case of restoration, the same adjective that was used with hindsight to emphasize the novelty of some forgotten theory should help keep that theory alive as an alternative to our modern theory.
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To complete this reconceptualization of the two theories as true alternatives as guided by the metaphor of the spectrum, I will reexpress the basic principle of each theory as a positive limit case of the same intermediate view. Then, while thinking the reexpressed basic principle of one theory, we will have yet a third kind of awareness of the basic principle of the other theory - we will know it to be 'on the other side' of some intermediate. Joining this new awareness to the other two, we will know our chosen principle to be at the extreme end of a band of different kinds of referents.
The intermediate principle I will make use of derives from a procedure actually introduced by Viète, although only within the context of his constitutive view. His method of syncrisis compares two correlated equations (similar equations having the same coefficients, but not necessarily the same signs) until the coefficients can be expressed in terms of the roots of the two equations. When the two correlated equations are contradictory or inverse, their respective roots are necessarily related as greater to lesser (since Viète is only considering positive real roots). When they are schizoid, one root is arbitrarily assumed to be greater than the other. Often Viète can go beyond this mere relative order and put these two roots into an absolute order by introducing other elements in continuous proportion to the roots as extremes, but this is not in general possible.
After turning this situation around somewhat so that the emphasis falls on a given equation instead of on a pair of equations, prohibiting the introduction of new elements as proportionals, and reconceptualizing in terms of referents, we have the following principle: The referents of a given equation are relatively ordered among themselves according to relations of greater & lesser; one of them is equal to the root of the given equation itself, while the others are roots of correlated equations. Note that the two characteristic features of a referent are here equally prominent. One is an obvious concomitant to the other, and neither one dominates. For now, I will call this the syncritic view.
One distinguishing feature of this view is that the referent elements, although relatively orderable, have among themselves the inexact size relations of greater & lesser only, and to this extent are not absolutely orderable. Whereas Viète, conceptualizing the syncritic view within the influence of the constitutive view, would have considered this feature to be due to a deficiency of the referent elements necessary to make up an order, I am making inexact size relations a natural part of the definition of this type of referent. If I now reconceptualize the constitutive view as a limit case of this aspect of the syncritic view, referent elements in the constitutive view can only be seen to be absolutely orderable in so far as they have perfectly exact relations of size among themselves, the exact relations in a ratio being a kind of limit case of the inexact relations of greater & lesser. The basic principle of the constitutive view thus becomes: The referents are absolutely ordered elements having exact relations, such as those in a continuous proportion. And on the other side of the syncritic view with its inexact relations, we dimly make out a view in which the referents have no relations at all, and hence no order either.
Again, since the referents are also roots, albeit not necessarily of the same equation, we can also reconceptualize the root-oriented view as a limit case on the other side of the syncritic view. Looking at the syncritic view from within the context of the root-oriented view, we would say that it expresses the coefficients of an equation in terms of the roots of the equation itself and correlated equations. But when the root-oriented view is taken as a limit case of the syncritic view, we would have to say that all the referents of a given equation are equal to roots of the same equation correlated with itself alone. And on the other side of the syncritic view? The outline of a theory in which an equation is correlated with equations so dissimilar to it that these other referents begin to appear to be extrinsic to the equation under consideration.
The syncritic view is thus only relatively alternative to the other two, since they are each limit cases of it. But to the extent that the root-oriented view and the constitutive view are limit cases of different fundamental characteristics of the syncritic view, we can say that they are true alternatives.
A sign that this reconceptualization is true in the sense that it is adequate to the dual nature of equations as expressed in the basic principle of each of the two theories - that is, that thinking the principle of one does not conceal the principle of the other, but rather gives an awareness of it - is the possibility that the one view could now be discovered within an investigation of the other one. Viète actually encountered the syncritic view within the constitutive view, and Girard himself criticized Viète's use of syncrisis (misunderstanding his intention) as not being directed toward finding all the solutions (positive, negative and imaginary) of a given equation. This path is made all the plainer, once the two theories have been reconceptualized in this way. For then the movement is from limit case to the case that it limits, then to a concomitant characteristic of this case, and then to the limit case of this characteristic. These movements are all within the scope of a focal logic.
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So the theory of equations has turned into a general investigation of the different kinds of referents that equations can have. I will conclude this editorial with the briefest sketch of how such a reconceptualized theory of equations could broaden out actual mathematical research into the properties of equations. First of all, there is the matter of the referents themselves. The sought root as well as the coefficients are more or less complex functions of referents; consequently, equations are theorems about referents. The referents themselves are potential counterparts to mathematical entities, and thus part of the algebraic calculus itself. The referents have relations of order among themselves; as a limit case they have no specified relations. The interrelatedness and quality of intrinsicness of the referents are roughly inversely proportional, like wave length and frequency, but this needs to be made more exact.
The different kinds of referents form a spectrum. A given theory of equations is distinquished by the selection of one kind of referent. The spectrum needs to be more fully defined. In a rough and ready way we already have the extremes, and one of the intermediate bands. But on closer scrutiny the bands themselves become articulated. For example, within the band defined by ordered and exactly related constitutive elements may be found absolutely ordered elements (as in a continuous proportion) or relatively ordered ones (as in a ordinary proporiton); the exactly related elements may belong to a geometrical proportion, or an arithmetical proportion, etc.
Each theory has a host of attendant concepts and procedures that must be singled out. For example, to relate the referents to the equation itself, the constitutive view uses ordinary algebraic argument on an unknown, the syncritic view uses the method of syncrisis, and the root-oriented view generates the equation by the multiplication of factors.
Then, there is a collection of theorems and problems about the properties of equations, each of which can be meaningfully posed in all the three individual subtheories that I have singled out. Being independent of any given subtheory, they could rightly be called the proper subject matter of our reconceptualized theory of equations. Although one theory might excel the others at untangling a given difficulty, each would have its own peculiar approach to these theorems and problems.
For example, the multiplicity of positive real solutions for certain equations can be studied by each of the three theories. For the root-oriented theory this is a true multiplicity and has to do with the generation of the equation by the multiplication of distinct factors. But under Viele's constitutive view this multiplicity is thought of as an essential ambiguity involved in relating the equation to its referential proportion. And in the syncritic view this property results from the fact that certain equations can be meaningfully correlated with others of exactly the same form, while others cannot.
Next, a prominent concept or conspicuous distinction in one subtheory may have analogues in the others, thus helping us uncover in them concepts that may be more remote and distinctions that may have been blurred. Or a comparison of the theories may help give a positive characterization to a negatively defined feature in one of them. This presupposes a general investigation of the relation of the constitutive elements, the syncritic elements, and the component elements for a given equation.
For example, when investigating the case of the cubic in which imaginaries appear in the general formula, Viète makes use of a correlated cubic equation in order to refer the given cubic to a continuous proportion of magnitudes that are all positive real quantities. (And the correlated equation is the ambiguous case of the cubic.) Or again, perhaps the idea of a group may turn out to be the way in which constitutive ideas manifest themselves in a theory in which the referents are assumed to have no order relations among themselves.
Finally there are all sorts of new subjects for investigation that are idiosyncratic to one theory or another.
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To RECAPITULATE: Quite early in its history, the algebraic theory of equations was confronted with a fundamental alternative. A crucial decision was made to accept this alternative, although not with full knowledge of what was at stake. This committed the theory to a certain direction, which later came to be accepted as the only correct way of proceeding. As the theory expanded and began to pile up successes, the original view was forgotten. Today, the very possibility that there ever was - or could be - a fundamentally different view seems incredible. The earlier way of proceeding has come to be seen as a dim groping for our own approach by a man ahead of his time.
It concerns me that there may be a similar tendency in the other sciences as well. To me, this would be a very insidious form of intellectual tyranny, something like an election in which there is only one candidate. The purpose of this editorial has been to present the modern mathematician with some more choices, and give him back a little of his freedom. Perhaps he will wish to assume a new intellectual habit. But if he still favors a root-oriented theory of equations, let it be a theory that he reaffirms.
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(The above essay was published along with "The Early Theory of Equasions - Treatises by Viète, Girard, & De Beaune". Below are selected entries from "A Glossary of Viète's Special Terms")
Constitution (constitutio). Implies moving to a more highly organized state by construction.
An equality and a proportion may both compare the same elements, but they do so in different ways. An equality introduces greater degree of conjunction of the elements while involving them in a single comparison of the simplest type. The corresponding proportion keeps the elements in greater distinctness while bringing them into a higher order of comparison, since a proportion is a comparison of ratios, and ratios are themselves comparisons more complex than equalities. So, as Viète says, a proportion is the constitution of an equality (that is, it sets the elements of the equality into relation with one another) while an equality is the resolution of a proportion (in the sense that it dissolves the higher order of comparison into a simpler).
In the case of an equation that reckons with species, the ultimate elements are roots. The equation itself is a statement predicating one species of another, either directly, or after altering the subject by some affirmation or denial. Although these species are made up of roots, the equation says nothing about the relations that these roots have to one another. When this equation is referred to a proportion, however, the roots are set into direct comparison with one another. Since roots belong to the realm of species, their relations are non-quantitative. What the proportion does is to set the roots into a certain order, and if the proportion is a continuous one, then the order is an absolute one, with every root having its position in the series. This is what it means to constitute an equation that compares magnitudes by means of species.
Viète uses systatical as a synonym for constitutive. Systatical is a Latinized form of the Greek word that Euclid uses for construction, as in the first problem of the Elements, where the problem is to construct an equilateral triangle. According to Aristotlean tradition, construction had to show the existence of a certain kind of figure - that its definition was free from contradiction. It accomplished this by displaying an example of the figure, a subject for the defining property. Viète could have done this with his equations, pointing to an actual geometrical situation to which the letters in the equation could be referred; this would be an exegetical theorem. And he occasionally does so in this treatise. But since the equation is about roots and species, he usually only refers them to a proportion; this suffices to show that the equation does not contradict itself by declaring impossible things about roots.
It is also useful to constitute equations to find more elegant solutions to problems. Since we are dealing with a more organized situation, it usually takes fewer operations to reveal what is sought in terms of what is known, although the solutions may be harder to discover.
Exegetical Theorem (exegeticum). Like a constitutive theorem, except it refers the roots compared in an equation to an actual geometrical situation in which certain lines, say, are proportional. Thus it goes one step beyond the constitutive theorem and invokes a subject to which the relation of roots can refer. Remember that the Greeks thought of construction as showing that the subject existed.
Viète conceived of an exegetical analysis as the final part of his analytic art, and wrote the necessary treatises to support it.
Species (Species). The counterparts of magnitudes in Viète' s algebraic logistic, or calculus. An equivalent Greek term is used by Diophantus in his Arithmetic.
Viète would probably have defined logic as the art of comparing things correctly. Different logics would compare things under different aspects. One way of comparing geometrical entities is from a quantitative point of view. Another might be from a figural point of view. Viète introduced still another logic that was interested in the formal aspects of magnitudes, which may roughly he defined as the role they play in a problem. For example, instead of considering a rectangle to be a figure or an area, it might he regarded as something two dimensional - a kind of function of its sides where the sides are present to one another in such a way that they create something of a higher order or genus. Viète calls this a plane or a factate; he calls the sides latitudes & longitudes. Alternatively, we might regard this rectangle as a homogeneum. This is to look at the rectangle as a kind of function of its substantial parts, which are present to one another in a conjunction of affirmation or denial. Thus, the parts are of the same genus as the whole. For example, the rectangle may he understood as the sum of two squares.
Viète's introduction of species was meant to direct attention to the ways in which magnitudes show themselves in algebraic comparison. This formal aspect is not something intrinsic to magnitudes like figure & size are. In fact, in the course of one and the same algebraic argument it is possible for a magnitude to change its role (see transformation). But Viète thinks that this is the proper way of regarding things when practicing the algebraic art.
A logic may he served by a logistic, which for Viète is the art of making vicarious comparisons of things. A logistic employs easily manipulable counterparts or proxies to exhibit the intended comparison of the things under examination - magnitudes, for instance. The numerical logistic of Viète's day compared things under the aspect of quantity, and it employed written numbers to exhibit this comparison. Viète's new logic also was served by a suitable logistic, and he called the elements of this logistic species.
Since there does not seem to be any one realm to which these functions intrinsically belong, and since even for magnitudes it is necessary to set up conventions defining how they are factates or homogenea, just about any written symbols could serve as the elements for this logistic. But it would be nice if we could find counterpart elements whose intrinsic relationships were not at variance with the very ideas behind factate or homogeneum functions. Ideally, they would even express these functions in some relatively uncontrived way.
At least two considerations would rule out written numbers. First of all, all numbers are homogeneous, while magnitudes are not. Again, every numerical homogeneum can ultimately be reexpressed as a sum of units, as a consequence of the fact that the unit elements of numbers are all equal. In the case of magnitudes, however, the homogeneum that is a conjunction by denial is not in general reducible to one that is a conjunction by affirmation, at least not in terms of the unequal but proportional roots that constitute the equation in recognition, which are the elements from which these functions must be made. Denial is an equally primitive operation.
Viète chose letters. From his formal point of view, letters can be elements in a factate or homogeneum function just as uncontrivedly as magnitudes themselves, because in different types of configurations letters can have a different import as sounds. A certain configuration of letters generates a syllable, which might be interpreted as a factate. A certain configuration of syllables (or occasionally individual letters) produces a word, which might be interpreted as a homogeneum. What Viète does is to take the principle behind syllable and word formation and refine it to the point where there is a perfect correspondence between the collections of letters, and the different types of factate and homogeneum functions that come up during the algebraic comparison of magnitudes.
The manipulable elements of a logistic have an ambivalent status as symbols. On the one hand, they are the tangible counterparts to things not readily compared directly. But, in so far as they are written on paper, they tend to be taken as signs pointing to a universal, much in the manner of any other written word. One is naturally led to wonder whether Viète thought that the species signified some pure and non-sensible entity, considering the philosophical importance of this word. It is hard to tell.
But I don't think that the introduction of species posed a new ontological problem for Viète. In the numerical logistic of the time, written numbers were not so much symbols meant to designate pure or ideal numbers, as easily manipulable examples of multitudes. The marks on paper were substitutes for the actual multitudes much as the beads on an abacus might serve as counterparts for a number of real things. The multitudes were implicit in the numerical figures themselves, the isolated figure 5, say, standing for a group of five single strokes of the pen. This is particularly clear in the case of Roman 'numerals', where the first three are simply one more stroke of the pen. The Arabic figures were introduced into the West merely as a better way of writing numbers, and preserved this sense of 'standing for' multitudes.
So the written collections of letters are species in the same sense that the numerical figures are numbers. At the very least we might say that they are each the privileged instances of the mathematical entities they exemplify.
Syncrisis (syncrisis). A good general word for bringing things into comparison. Theon of Smytna, for example, says that homogeneous or homospecious things are called terms when they are taken into syncrisis.
Suppose we should want to refer the coefficients or an equation to roots that are all at the same time roots of the equation, somewhat in the manner of the modern root-oriented theory of equations. For Viète, this would in general be impossible, since not every equation is ambiguous, and so there would not usually be even two roots. The proportion that constitutes the equation must necessarily contain roots that are independent of the roots that solve the equation. But it is in general possible to refer the coefficients to roots that are either roots of the equation under consideration, or else roots of correlated equations. Syncrisis is the method that shows us how to do this.
Viète uses syncrisis as an intermediate stage in the referring of an equation to a continuous proportion. By making the coefficients of the proposed equation all depend solely on the roots of the two correlated equations, syncrisis moves in the general direction of constitution, because it brings the two roots into a relation that was not stated in the original equations themselves. But this only allows the two roots to be taken in a relation of prior to posterior. It does not assign the roots to an order in a series of continuous proportionals. There is no way of knowing how many roots separate them, so we do not know whether they are first and second, or first and third, or what. That is, syncrisis is not constitution in the fuller sense of moving to the higher order of comparison found in a continuous proportion. Viète appends a series of theorems on proportions that suffices to complete the referral of the correlated equations to a proportion.
Systatical Theorem (systaticum) . Used as a synonym for constitutive. This is derived from the Greek word that Euclid usually uses when he states the problem of constructing something.
Transformation (transmutatio). Whereas the four basic species operations result in the introduction of new species, they leave the species of the subject magnitudes unchanged. However, when an altered root is conceded to be equal to a new one, the original root changes its significance - in Viète's terms, it assumes a new species. For example, when one root or side is led into another, and the factate made from these is conceded to be equal to some other plane root, then one of these is changed into a longitude and the other into a latitude. The letters themselves have changed their significance.
Zetesis (zetesis). Algebraic argumentation that begins with a hypothesis about the magnitude in question and leads to an equation. Then, setting this equation in order. From a common Greek verb meaning 'to seek'. Doubtless suggested by one of the two types of analysis elaborated by Pappus in his general description.