Geometric constructions in Classical and Hellenic period

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Geometric constructions in Classical and Hel-lenic period

If we take a wide look at the history of mathematics, we find two great traditions of operative approaches: the “geometric” (or constructive) and the “algebraic” (or computational) 1. If today the mainstream vision of mathematics is algebraic, the historically most lasting perspective was the geometric one, mainly thanks to Euclid’s Elements 2.

Compass and straightedge constructions

Euclid’s plane geometry constructions involve the use of lines and circles to recursively generate points in their intersections. According to the practical drawing of lines and circles on a sheet, this construction are usually called “compass and straightedge constructions,” and are probably the idealization of the ancient “peg and cord” constructions. The reason for the choice of such tools was lost in the past, but many reconstructions have emerged ever since (about philosophical, epistemological, and religious standpoints 3). More concretely, there are also technical reasons. In fact, these constructions were particularly useful because they were quite precise in practice and general in application 4.
Thus, the theory developed in Euclid’s Elements provides a mathematical model of the activities available with these tools, a model well set in rigorous scientific canons: In particular, the existence of geometric objects is provided by their constructability.
The constructive power of these tools is captured by the following axioms: (R) Given two distinct points A, B, it is possible to construct the line through A and B.
(C) Given two distinct points A, B, it is possible to construct the circle with center A and radius AB.
Given any two (distinct) elements that intersect, their points of intersection are tacitly supposed to be constructed as well. The straightedge is not allowed to be marked, and the compass is not allowed to be used as a divider. That is, compasses have to be set according to postulate (C) every time they are used. The second and third propositions of the first book of the Elements show that the straightedge and “classical compass” can be used to simulate what is sometimes called the “modern compass,” which can perform the following operation that is more general: (MC) Given two distinct points B, C, it is possible to construct the circle with center A and radius congruent to BC. Thus, the adoption of a “modern compass” in place of a “classical one,” even though simplifying some constructions, does not extend the class of solvable problems. As I am going to note, other generalizations of the allowed tools will imply an extension of constructions.

Neusis constructions

Classical Greek geometry recognized that certain problems, such as doubling a cube, trisecting an angle, squaring a circle, and constructing certain regular polygons, did not appear to be possible using a compass and an unmarked straightedge alone 5. The problems themselves, however, are solvable, and the Greeks knew how to solve them, without the constraint of working only with straightedge and compass.
Archimedes knew 6 that the addition of two marks on the straightedge was enough to make the trisection of the angle and duplication of the cube possible. The classical Greek literature provides several other examples of tools permitting otherwise impossible constructions. The majority of them, such as the marked straightedge, permitted the construction of cube roots, hence the solution of all cubics (and quartics). Others were more powerful; the Archimedean spiral permits the n-section of any angle 7 and thus the construction of any polygon; and the quadratrix of Hippias allows the circle to be squared. In this subsection, eguali con dati numerici diversi. [. . . ] L’efficienza dell’algebra geometrica basata sulla riga e il compasso era strettamente connessa alla possibilità di effettuare precisi disegni su fogli di papiro.” Russo [2001].
Figure 2.1: Consider a point O, a line r and a distance d. On a generic line through O that intersects r at P consider the points Q1, Q2 which are distant d from P . The loci of the points Q1, Q2 are two “conchoids.”
I will focus on the extension of Euclid’s tools with the marked straightedge, the so-called neusis constructions.
For example, the use of a markable ruler permits the following construction. Given a line segment, two lines, and a point, one can draw a line that passes through the given point and intersects both lines so that the distance between the points of intersection equals the given segment. The Greeks called this “neusis” because the new line tends to the point 8. In this expanded scheme, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. It follows that, if markable rulers and neusis are permitted, the trisection of the angle and the duplication of the cube can be achieved; the quadrature of the circle is still impossible. Moreover, using such tools it is possible to draw curves that are different from lines and circles: As evident from Fig. 2.1, neusis is at the core of the con-struction of the conchoid of Nicomedes. Nicomedes, like many geometers of the third century B.C., tried to solve the problems of doubling the cube and trisect-ing the angle, whereby he created the conchoid. If one allows the extension of geometric constructions not through an extension of the constructive postulates but through the introduction of new curves, the conchoid allows solving the duplication of the cube and the trisection of the angle without the introduction of the neusis.
Hence, Euclid’s limits can be overcome with other tools. However, when can the new methods be considered acceptable in geometry? In general, what can be considered a legitimate geometric construction? These complex questions are now said to be about “geometrical exactness.” With a chronological jump, the need of a new canon for constructions became even more relevant when algebraic tools provided a different way of problem solving, an analytical method requiring a legitimation in a geometric paradigm.

Geometric exactness in the early modern period

In this section, I introduce the early modern concern for “exactness” 9 of geometrical constructions, evincing the pivotal role of Descartes’s La Géométrie.

Bos’s perspective

Amongst historians of mathematics, Henk Bos’s works on Cartesian ana-lytical geometry and Leibnizian calculus have provided the starting point for all those who have tried to understand the conceptual developments of mathe-matics in the crucial period between the Renaissance and the Enlightenment. 10 I can especially use Bos’s interpretation of early modern mathematics as the starting point of this thesis.
The peculiarity of Bos [2001] is the perspective. Often problems with re-spect to the changes in concepts of geometric constructions, typical of the early modern period, are looked from the later perspective of mature analytical geom-etry. Contrarily, Bos focuses on a set of problems that were of high significance for mathematicians in that period of mathematical revolutions. These math-ematicians were troubled by the following questions: Which constructions can be considered legitimate? Which ones are simpler? Regarding mathematical entities, when can they be considered totally achieved? What does it mean for a problem to be solved and its solutions to be found?
In this context, the concepts of exactness, certitude, and precision were fre-quently used and discussed because of the conceptual and methodological prob-lems owing to the introduction of algebra as an analytical tool for the solution of geometric problems: How can algebraic solutions be considered legitimate inside of a universally accepted geometric paradigm? The geometric interpre-tation of the new algebraic techniques posed enormous problems, as evident in the effort that, even if in different ways, mainly Viète and Descartes spent on the problem. All these topics totally disappeared in the 18th century because of the general affirmation of symbolic procedures as autonomous from geome-try. Even if historically forgotten, the reflection on such topics is central to the understanding of the evolution of mathematics in that revolutionary period.
In Bos’s book, the main role is played by Descartes, that is analyzed with in mind these early modern questions about constructions. From this perspective, in contrast to the future interpretation of reducing the study of curves to their defining equations, Bos supports that for Descartes the equation is just a part of the definition of a curve (analysis), because, in order to practice geometry, one had to produce the geometrical construction (synthesis). Thus, even introducing the power of algebra into geometry, it refers only to the analytical part, while the synthetic counterpart is still necessary. From this perspective, Descartes did not depart from the ancient view of considering a solution known only if constructed out of geometrical elements.
My aim is to adhere to the ancient paradigm of geometric constructions, and to extend it to differential objects along a direction different from the one that historically became dominant (the introduction of infinitesimal entities in analysis).

Defining the exactness problem

In this subsection, I take some useful concepts from the “General Introduc-tion” of [Bos, 2001, pp. 3–22]. The “exactness” of mathematics is an evolving idea. It was especially really fluid in the period between the Renaissance and the Enlightenment. As observed at the beginning of the chapter, the problem of exactness in geometric constructions was present at least since the classical era, specially to face problems not easily solvable just with Euclid’s lines and circles. Though soon after its publication, Cartesian geometry became a widely ac-cepted canon, prior to Descartes’s criterion, there had been many other attempts of exactness in the early modern period. The leitmotiv of such attempts was the need to answer the foundational problems given by the introduction of algebra as an analytical tool for the solution of geometric problems. In fact, a solution was acceptable if it could be justified by a suitable geometric interpretation. It was thus necessary to define the acceptability of solutions and constructions in the geometric paradigm. From the 16th to the 18th centuries, many mathe-maticians asked themselves, in the new context, what it meant for a problem to be “solved” or for a mathematical object to be “known.” According to clas-sical Greek geometry, these questions were both answered by the acceptance of a canon for geometric constructions, usually by straight lines and circles con-structions. However, in the early modern period, algebra strength in problem solving suggested overcoming the classical means of construction to interpret its solutions geometrically: There was a need for a new canon of acceptable procedures.
Since the classical times, Euclid’s geometry was extended by families of curves out of lines and circles 11. The introduction of algebra as a tool for geometric problem solving caused the growth of the constructible curves 12, so the question of when a curve was sufficiently known, or how it could acceptably be constructed, acquired a new urgency. Bos calls “representation of curves” the descriptions of curves that were considered to be sufficiently informative to make the curves known. For representing curves, mathematicians resorted to the means which geometry offered for making objects known—the conceptual apparatus of “construction.”
Therefore, the exactness problem became the problem of choosing acceptable means of construction. Even if this choice was justified by a meta-mathematical argument (similar to the choice of axioms in a theory), the reasons for or against accepting procedures of construction were very important in the development of mathematical practice: they determined directions in mathematical research, and they reflected the mental images that mathematicians had of the objects they studied. To justify which procedures were acceptable, mathematicians had to explain, to themselves or to others, what requirements would make mathematical procedures exact in the above sense. Thus, they had to inter-pret what it means to proceed exactly in mathematics. Bos calls this activity the “interpretation of exactness,” and suggests the following cases of basic at-titudes: appeal to authority and tradition; idealization of practical methods; philosophical analysis of the geometrical intuition; appreciation of the resulting mathematics; refusal/rejection of any rules, or non-interest. The most influ-ential cases were probably the “philosophical analysis of geometrical intuition” (Descartes’s approach), which required a cognitive attention on how geometric intuition can be transformed in acceptable procedures, and the lack of inter-est (Leibniz’ approach), in which, as in the dominant modern mathematical standpoint, procedures are justified by their utility in problem solving.
The structure of the story of construction and representation in early modern mathematics is basically simple. It comprises two slightly overlapping periods, c. 1590–c. 1650, c. 1635–c. 1750, and one central figure, Descartes. During the first period, questions about construction arose primarily in connection with geometrical problems that required a point or a line segment to be constructed and admitted one or at the most a finite number of solutions (e.g. dividing an angle in two equal parts, finding two mean proportionals between two given line segments). If translated into algebra, problems of this type led to equations in one unknown. Around these problems a considerable field of mathemati-cal activity developed, which may be considered as the early modern tradition of geometrical problem solving. Indeed, the adoption of algebraic methods of analysis provided the principal dynamics of the developments in the field.
In Bos’s opinion Descartes’s La Géométrie of 1637 derived its structure and program from this field of geometrical problem solving. The two main themes of Descartes’s book were the use of algebra in geometry and the choice of appro-priate means of construction. The approach to geometrical construction that he formulated soon eclipsed all other answers to the question of how to con-struct in geometry. Thus, Descartes closed the first episode of the early modern story of construction by canonizing one special approach to the interpretation of exactness concerning geometrical constructions.
Nevertheless, La Géométrie may also be seen as the opening of a second period lasting until around 1750. In this period, the problems that gave rise to questions about construction and representation were primarily quadratures and inverse tangent problems. These belong to a class of problems in which it is required to find or construct a curve. If translated in terms of algebra, these problems lead to equations in two unknowns, either ordinary (finite) equations or differential equations. It was from this field that, in the period 1650–1750, infinitesimal analysis gradually emancipated itself as a separate mathemati-cal discipline, independent of the geometrical imagery of coordinates, curves, quadratures, and tangents, and with its own subject matter, namely, analytical expressions and, later, functions. This process of emancipation, which might be called the “de-geometrization of analysis,” constituted the principal dynamics within the area of mathematical activities around the investigation of curves by means of finite and infinitesimal analysis. It was strongly interrelated with the changing ideas on the interpretation of exactness with respect to construction and representation.
Although the interpretation of exactness with respect to geometrical con-struction and representation was discussed with some intensity during the early modern period, no ultimately convincing canon of geometric constructions was found to face the problems of the second period. By 1750 most mathematicians had lost interest in issues of geometrical exactness and construction; they found themselves working in the expanding field of infinitesimal analysis, which had by then outgrown its dependence of geometrical imagery and legitimation.
These changes were brought about by such processes as the habituation to new mathematical concepts and material, and the progressive shift of method-ological restrictions. By habituation, a mathematical entity that was earlier seen as problematic (such as some transcendental curve) could later serve as solution of a problem, even though the mathematical knowledge about it had not changed essentially. Methodological restrictions were mitigated or lifted as the result of conflicts around the legitimacy of procedures and because of the appeal of new mathematical material.

Analysis and synthesis in La Géométrie

Early modern exactness problem dealt with the definition of appropriate norms for deciding if some objects, procedures or arguments can or cannot be considered geometrical. All the various attempts in this direction had a minimal common basis, given by Euclid’s plane geometry, suitably extended. I previously said that Descartes provided a widely accepted canon of geometrical construc-tion: His strength was given by his perspective of a “philosophical analysis of geometric intuition.” In fact, I have to remind that, quoting Bos:
“The Geometry 13 served as an illustrative essay accompanying the Discourse on the method. Descartes did not explicitly discuss the links between the method of the Geometry and the general rules of methodical thinking expounded in the Discourse. Yet, for instance, the second and third of the four rules expounded in Part 2 of the Discourse 14 might easily be seen as exemplified by the procedures of analysis and synthesis, respectively, as detailed in the Geometry.
Indeed the method of the Geometry consisted of:
• An analytic part, using algebra to reduce any problem to an appropriate equation;
• A synthetic part, finding the appropriate construction of the problem on the basis of the equation.”
The construction of a curve had to be obtained as the simplest possible: This simplicity should be achieved reducing the geometrical problem to an algebraic equation (in one unknown) of lowest possible degree, later transformed in a certain standard form. This algebraic part, even if essential, was just one of the two parts of the method: “[t]he fact that algebra does not provide geometrical constructions merits emphasis because too often Descartes’s contribution to geom-etry is presented as the brilliant removal of cumbersome geometrical procedures by simply applying algebra. In fact, algebra could only do half of the business, it could provide the analysis and reduce problems to equations. The other half of the job, the synthesis, the geometrical construction of the roots of the equations, remained to be done.
The synthetic part of Descartes’s program presented the most pro-found questions. They concerned the conception of geometrical con-struction itself, in other words the interpretation of constructional exactness. That interpretation required a demarcation of the class of curves acceptable for use in constructions and a criterion to judge the simplicity of these curves [. . . ]: acceptable curves were traced by acceptable motions; they were precisely those that had algebraic equations; they were simpler in as much as their degree was lower.” 16
With regard to Descartes’s main purpose in geometry, I propose two inter-pretations: the first one is that his purpose was to provide a general method for geometrical problem solving (Bos [2001]), while the second one (discussed in detail in the next subsection) is that the origin was to get a “conservative extension” of Euclid’s geometry (Panza [2011]). These visions are not mutually exclusive, and for my standpoint it is not so important to consider one of them most basilar than the other. Quoting Bos: “By 1635 [. . . ] the first generation of mathematicians active in the early modern tradition of geometrical problem solving had passed away. In their time the major innovation in the field was Viète’s use of his new algebra 17. Some mathematicians, Clavius , for instance, paid no attention to this innovation; Kepler even rejected the use of algebra in geometry. But it seems that by 1635 the practice of geometrical problem solving without algebra [. . . ] had vanished from the scene of active mathematical investigation.” 18
After that, regarding Descartes’s La Géométrie: “The core of its influence consisted in the spread of Descartes’s in-sights and techniques about the relation between curves and their equations or, more generally, about the interplay between figures and formulas. [. . . ] My analysis of the Geometry in the preceding chap-ters has shown, however, that Descartes’s main motivation in writing the book was not to expose the equivalence of curve and equation. Rather, it was to provide an exact, complete method for solving “all the problems of geometry.” [. . . ] Thus, [. . . ] the main influence of the book did not concur with its program. Indeed the Geometry exerted its main influence despite its primary motivation.” 19

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Cartesian canon of constructions

Even if Panza [2011] agrees with the importance of problem solving in Descartes’s program, he goes further and proposes that “Descartes’s primary purpose in geometry appears to be a foundational one, and his addressing the exactness concern appears as a crucial ingredient of this purpose” 20, namely that of obtaining a “conservative extension” of Euclid’s plane geometry. Panza bases his reconstruction on an analysis of the ontology of Euclid’s geometry. In contrast to modern mathematical theories, Euclid’s geometrical ontology “is composed of objects available within this system, rather than objects that are required or purported to exist by force of the assumptions that this system is based on and of the results proved within it” 21. These objects to be avail-able have to be constructed. Euclid’s constructions require that appropriate diagrams be drawn, and these constructions are just “procedures for drawing diagrams in a licensed way, to the effect that an EPG 22 problem is solved when appropriate diagrams, representing some objects falling under the concepts this problem is concerned with, are so drawn, or imagined to have been drawn” 23. To trace curves beyond straight lines and circles, it is fundamental to define the role that instruments have in “diagrammatic constructions.” More precisely, these instruments can be constructively used on a plane in two ways: “either in the tracing way, i.e., by making them trace a curve; or in the pointing way, i.e., by making them indicate some points (which are then taken to be obtained) under the condition that some of their elements coincide with some given geometrical objects, or meet some other conditions relative to given objects. If an instrument is used in the former way, once a curve is traced, it can be put away, and this curve taken as constructed. If it is used in the latter way, the sought-after points can only be indicated by appropriate elements of it. [. . . ] This suggests two different sorts of constructive clauses, licensing respectively obtaining curves by tracing them through instruments, and obtaining points by using instruments in the pointing way.” 24
The latter use implies that one can move (parts of) the instruments “until they reach a position that satisfies a coincidence condition relative to other diagrams representing some given geometrical objects” 25. This is a use of diagrams es-sentially different from Euclid’s, where coincidences are not acknowledged by inspecting moving diagrams but imposed on fixed diagrams by drawing them. Descartes’s geometry is today usually considered as the beginning of modern mathematics, because of the revolutionary possibility of describing and analyz-ing classes of geometrical curves through equations. However, here I want to focus on the genetic relation that Cartesian geometry has with classical one. In particular: “EPG is often described as dealing with ideal and immutable self-standing objects or forms, which we can only inaccurately depict. If EPG were so understood, the use of instruments in geometry (both in the pointing and in the tracing way), and more generally the appeal to motion, should be considered as entirely extraneous to its spirit, unless they were merely seen as tricks for achieving convenient depictions of ideal forms. The situation is different if it is granted that EPG objects are obtained through diagrammatic constructions. It then becomes natural to consider the admission of new procedures for drawing diagrams, also by using instruments, as a proper way of conservatively extending EPG.
In classical geometry, the use of instruments to obtain geometrical objects did not go together with fixing precise conditions that such a use of an instrument had to submit to. As a matter of fact, this made the exactness norms of geometric objects inaccurate and contributed highly to the fluidity of classical geometry.” 26
This gives the motivation of Descartes’s foundational program. In contrast to the other attempts before him, Descartes’s geometry is a closed system with an ontology composed of objects available within it through precisely defined diagrammatic constructions: These well-framed boundaries can be seen as a conservative extension of Euclid’s geometry. However, I still have to be precise about the admissible instruments and to justify their acceptance.
In the La Géométrie Descartes criticized the “ancients” for having termed “mechanical” any curves other than circles and conics, because also circles and straight lines “cannot be described on a paper without the use of a compass and a ruler, which may also be termed instruments” 27. According to the previously introduced terminology, Descartes excluded from geometry the use of instru-ments in the “pointing way,” according to an interpretation of diagrammatic construction coherent with Euclid’s one. So the search for exactness norms is reduced to the identification of an appropriate class of instruments (later de-noted “geometrical linkages” 28) that, when used in the tracing way, trace curves that are admitted in geometry just because they can be so traced (Descartes named these curves “geometrical”).
Regarding these instruments, Descartes did not precisely define geometrical linkages, but, in a more or less explicit and general way, he put some require-ments that such machines have to satisfy. Even though I will not enter in the problems of suitably defining acceptable geometrical linkages, I have to cite that, according to [Panza, 2011, section 3.2], it is possible to characterize “geometri-cal” curves as objects obtained by ruler, compass and reiteration. Strengthening the connection between Descartes’s and Euclid’s canons, this perspective focuses on the way in which the first one is an extension of the second.

Beyond Cartesian tools

With respect to the consequences of Cartesian geometry, Bos asserted that: “fairly soon after Descartes’s Geometry mathematicians were so far habituated to algebraic curves that the equation of such a curve no longer presented a problem (how to construct the curve with that equation); rather it represented an object (the curve with that equation).
The habituation to non-algebraic curves took more time. This was partly because the representation of such curves was far from triv-ial; there were (at least until c. 1700) very few notational means available to express their equations. In the absence of analytical means of representation, a non-algebraic curve could only be imag-ined and talked or written about in terms of a geometrical procedure to construct or trace it. In the case of non-algebraic curves, these procedures involved combinations of motions, or pointwise construc-tions, which Descartes had expressly banned from genuine geometry because, in the case of non-algebraic curves, they did not provide proper knowledge of the objects.
A number of mathematicians felt that a reinterpretation of geometri-cal exactness was needed, overcoming the obstacle of the restrictive Cartesian orthodoxy. Thus, in the second half of the seventeenth century, Descartes’s ideas about genuine geometrical knowledge in-duced a new debate on the interpretation of exactness in connection with the proper representation of non-algebraic curves.” 29
Even though with some exceptions (specially in Great Britain), soon after the geometrical revolution of Descartes, it was suddenly accepted the analyt-ical part of its program (a well-framed introduction of algebra in geometry), while the interest in geometric constructions remained alive just to justify tran-scendental curves (not treatable with polynomial algebra). Especially, even if non-algebraic curves were well known by Descartes (examples of mechanical curves included the quadratrix, the Archimedean spiral, the cycloid), it was the “inverse tangent problem” that generated a wide class of curves for which Descartes’s tools were not powerful enough. Therefore, it was time to overcome Cartesian canons through an extension of the allowed “tracing machines” to per-petuate the paradigm of geometrical constructions (there was the acceptance of a kind of motion considered non-geometric by Cartesian canon, the “tractional” one). However, if Descartes’s reasoning was oriented to a closed class of con-structible objects, the new attempt was much more oriented to mathematical freedom. In this vision it is important the role of Leibniz, that hardly opposed to Cartesian restrictions 30. Behind the collapse of the geometric paradigm in front of the power of the analytical counterpart there was the passage from “finite” to “infinitary” entities 31, unreachable with finite instruments of dia-grammatic constructions. In this thesis I suggest to exhume the paradigm of geometric constructions in order to avoid the use of entities and procedures more or less implicitly recalling the infinity, so to finitely extend the balance between machines, geometry and algebra beyond Descartes.

Table of contents :

1 Preface 
1.1 Exactness of constructions
1.2 The role of machines
1.3 Schema of the work
2 Historical introduction 
2.1 Geometric constructions in Classical and Hellenic period
2.1.1 Compass and straightedge constructions
2.1.2 Neusis constructions
2.2 Geometric exactness in the early modern period
2.2.1 Bos’s perspective
2.2.2 Defining the exactness problem
2.2.3 Analysis and synthesis in La Géométrie
2.2.4 Cartesian canon of constructions
2.3 Beyond Cartesian tools
2.3.1 A brief history of Tractional motion
2.3.2 Leibniz’s criticism of Descartes
2.3.3 Vincenzo Riccati’s theory of geometric integration
2.3.4 Changes of paradigm: Geometry, algebra, use of infinity
2.3.5 A note on computation
3 From Euclid to Descartes 
3.1 Mathematical modeling: A behavioral approach
3.1.1 The universum and the behavior
3.1.2 Behavioral equations
3.1.3 Manifest and latent variables
3.2 Classical machines
3.2.1 Primitive objects of Euclid’s geometry
3.2.2 Components of classical machines
3.2.3 Construction rules for classical machines
3.2.4 Characterizing ruler and compass constructions
3.2.5 Defining equivalence between classical machines
3.2.6 The role of the cart in classical machines
3.3 Algebraic machines
3.3.1 Extending classical machines: Neusis constructions
3.3.2 Machine-based approach
3.3.3 Behavioral approach for algebraic machines
3.3.4 Arithmetic operations with algebraic machines
3.3.5 Real algebraic geometry background
3.3.6 The full behavior is a real algebraic set
3.3.7 Any real semi-algebraic set is an external behavior
3.3.8 Equality between algebraic machines
3.4 Notes on other algebraic constructions
3.4.1 Machines with strings
3.4.2 Curves constructed as ruler-and-compass loci
3.4.3 The role of the cart in algebraic machines
4 Differential machines 
4.1 Machines beyond algebraic ones
4.1.1 Tangent problems for algebraic curves
4.1.2 Dynamical slope field with algebraic machines
4.1.3 Tractional extension of machines
4.1.4 Defining differential machines
4.2 Setting differential machines
4.2.1 Definition of the universum
4.2.2 Full behavior as solution of differential polynomial systems
4.2.3 Differential systems are solved by differential machines
4.2.4 First example and note on “independentization”
4.2.5 Note on initial conditions
4.2.6 Role of the cart in differential machines
4.3 Analytical tools
4.3.1 Brief history of differential algebra
4.3.2 Differential algebra
4.3.3 Differential elimination
4.3.4 Solved and unsolved problems
4.4 Problem solving
4.4.1 The example of the cycloid
4.4.2 External behaviors and constructible functions
4.4.3 Equivalence between differential machines
4.4.4 Differential machines equivalent to algebraic ones
4.4.5 Conclusive notes
5 Complex machines 
5.1 Solving complex problems
5.1.1 Complex functions representation
5.1.2 From real to complex differential polynomials
5.1.3 Some remarks
5.1.4 A machine for the complex exponential
5.2 Some properties of the pivot point
5.2.1 Introduction of the pivot point
5.2.2 Tangents at output points in function of the pivot
5.2.3 Planar kinematics
6 Differential machines as physical devices 
6.1 Differential machines and integraphs
6.1.1 Integraphs with only straight components
6.1.2 Integraphs with curved guide
6.1.3 Integraphs with curved ruler
6.1.4 Integraphs respecting tangent conditions
6.2 The logarithmic compass
6.2.1 Introducing the device
6.2.2 Logarithmic compass extend ruler and compass
6.2.3 Applications to two classical problems
6.2.4 Open questions
6.3 Applications in math education
6.3.1 Re-structuration of calculus
6.3.2 Artifacts in math education
6.3.3 The tangentograph
6.3.4 A new concrete differential machine
7 Conclusions and future perspectives 
7.1 Balance between machines, algebra and geometry
7.1.1 A conservative extension of Descartes’s canon
7.1.2 A new dualism beyond polynomial algebra
7.1.3 Beyond differential machines
7.2 Foundational reflections on calculus
7.2.1 Cognitive approach
7.2.2 Computational approach
7.2.3 Toward a definition of exactness
7.3 Open problems and perspectives
Bibliography

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