Thesis Chapters by Davide Crippa
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Papers by Davide Crippa
History and Epistemology in Mathematics Education - Proceedings of the 9th EUROPEAN SUMMER UNIVERSITY, 2023
In this paper, based on a workshop held at ESU 2022 (Salerno), aimed at secondary school and univ... more In this paper, based on a workshop held at ESU 2022 (Salerno), aimed at secondary school and university teachers, as well as historians and didacticians, we suggest a new approach to calculus from a constructive geometric perspective. Specifically, we will provide a historical presentation of certain geometric instruments related to calculus, and we will introduce a new device for hands-on experiences. After that, we will describe the activities of the workshop presented at the latest ESU 9 in Salerno, in which we alternated the introduction of historical sources and laboratory activities.
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Vorträge des XI. Internationalen Leibniz-Kongresses 31. Juli – 4. August 2023, Leibniz Universität Hannover, Deutschland, 2023
Leibniz and the impossibility of squaring the hyperbola
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Almagest, 2022
From the middle of the 18th century on, the central authority in Vienna strove to increase its co... more From the middle of the 18th century on, the central authority in Vienna strove to increase its control over education in the Habsburg lands, attempting to introduce a series of reforms in university education in order to modernize the curricula in lower and higher educational institutions, institutions in which the influence of the Jesuits was predominant until the Society of Jesus’s suppression by the Pope in 1773. A consequence of these reforms was that the teaching of mathematics changed significantly in Habsburg territories between circa 1750 and 1784.
In this paper, we shall survey the content and structure of the course in Elementary Mathematics at the Charles-Ferdinand University in Prague. The teaching of mathematics at this college, which is fairly well documented, represents a fruitful case study for assessing the circulation of modern ideas in mathematical teaching. These ideas were promoted by contemporary authors like Christian Wolff or Nicolas Louis de la Caille, and were assimilated by the local scholarly community.
Since the present study represents no more than a preliminary exploration, our survey will cover only the period running from circa 1750 up to the year 1784. The terminus a quo here marks the publication of the first compendia written by a Jesuit professor from Prague and the terminus ad quem marks the end of Latin as the language of higher education. The switch to German as the official language of teaching introduced important changes, such as the adoption of new German textbooks and the end of Jesuit cultural hegemony even in scientific writings.
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Historia Mathematica, 2021
In 1804 the chair of Elementary Mathematics at Prague University became vacant and a selection pr... more In 1804 the chair of Elementary Mathematics at Prague University became vacant and a selection procedure, which consisted of a written and an oral examination, was announced. Only Bernard Bolzano and Ladislav Jandera took part in it. Jandera was appointed to the chair, whereas Bolzano became Professor of “Religious Doctrine.” In this paper we examine the context of this Concursprüfung, the performance of both candidates and the outcome, based on a number of related papers preserved at the Czech National Archives. In addition to this, we publish for the first time, albeit only in the online version of this paper, the transcription and English translation of Bolzano's written examination.
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Boston Studies in the Philosophy and History of Science, 2019
In this paper, I shall discuss Leibniz’s considerations on the problem of exactness, as they can... more In this paper, I shall discuss Leibniz’s considerations on the problem of exactness, as they can be reconstructed from published and unpublished letters, notes, drafts and sketches he did while Paris between 1673 and 1676. Leibniz’s critical target was Descartes’ Géométrie (1637). Descartes had managed to include into geometry all algebraic curves on the ground that they can be constructed by one continuous motions. Few rogue elements, called “mechanical curves” did not comply with this criterion of exactness, and therefore were not allowed in solving geometrical problems: these were, for example, the Archimedean spiral, the quadratrix and the cycloid. From the beginning of his mathematical studies, in 1673, and throughout his career, Leibniz set the task to reform the Cartesian criterion of exactness. Leibniz objected that Descartes’ criterion is not tenable, because there exist curves constructible by one continuous motion which cannot be associated to any algebraic equation. In this paper, I shall discuss several examples of such geometrical, non-algebraic curves: evolutes and involutes, the parabolic trochoid (the curve generated by the rolling of a parabola along one of its tangents) and the cycloid. The key to reform the Cartesian demarcation was, for Leibniz, to admit within geometry curve constructions obtained by strings that could be bent into
arcs. In this way, the previously mentioned curves resulted geometrical insofar as they could be generated by one and continuous motion.
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British Journal for the History of Mathematics , 2019
In this paper, I shall reconstruct the stay in Italy of James Gregory (1638–1675), Regius profess... more In this paper, I shall reconstruct the stay in Italy of James Gregory (1638–1675), Regius professor of mathematics at St Andrews. According to a standard account, Gregory spent four years (1664–1668) in Padua, as Stephano degli Angeli's student. However, this claim is problematic. First, Gregory's stay in Padua is confirmed only for the years 1667–1668. Second, the existence of a partial scribal copy of Vera quadratura circuli, ellipseos et hyperbolae in sua propria specie inventa et demonstrata, Gregory's debut work in the domain of quadrature problems, as well as a number of letters preserved at the National Library of Florence, suggest that relations between Gregory and Italian mathematicians were more complex and varied than have been suspected. On the basis of new, albeit scarce, textual evidence, I shall advance a few conjectures regarding scholars and philosophers that Gregory could have met in Padua, Rome and perhaps Florence.
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The Mathematical Intelligencer, 2019
After a brief historical introduction of geometric constructions of transcendental curves, we pre... more After a brief historical introduction of geometric constructions of transcendental curves, we present a new ideal machine that constructs both the tractrix and the logarithmic curve by virtue of the same motion.
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Revista Portuguesa de Filosofia, 2017
In this paper, we explore the role of the theory of proportions in the constitution of Cartesian ... more In this paper, we explore the role of the theory of proportions in the constitution of Cartesian geometry. Particularly, we intend to show that Descartes used it in an essential way to achieve a unification between geometry and arithmetic. Such a unification occurred mainly
by redefining the operation of multiplication in order to include both operations among segments and among numbers. Finally, we question about the significance of Descartes’ algebraic thought. Although the goal of Descartes’ Géométrie is to solve geometric problems, his first readers emphasized the role of algebra as a study of relations.
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In this paper, I shall study the last proposition of the De Quadratura Arith-metica (1676) of G.W... more In this paper, I shall study the last proposition of the De Quadratura Arith-metica (1676) of G.W. Leibniz, where we find a proof that it is impossible to solve algebraically the quadrature of an arbitrary sector of the circle, the hyperbola and the ellipse. I shall deal with the quadrature of the circle only, and I shall put this mathematical result into its proper context, that is to say the controversy that had opposed James Gregory and Christiaan Huygens several years earlier. Probably under Huygens' guidance , Leibniz studied the documents related to the controversy and wrote interesting observations about it. These observations show that Leibniz was led to investigate the issue of impossibility with the hope of correcting some flaws in Gregory's reasoning, and eventually come up with an original impossibility theorem.
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Drafts by Davide Crippa
Forthcoming. Please, do not quote or cite without permission.
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Leibniz on the construction of curves by strings. Please, do not quote or cite without permission.
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Conference Presentations by Davide Crippa
In this workshop, aimed at high school and university teachers, as well as scholars in history an... more In this workshop, aimed at high school and university teachers, as well as scholars in history and didactic of mathematics, we will suggest new ways to approach calculus from a constructive geometric perspective, using both historical materials and laboratory experiences through the manipulation of artifacts.
The workshop will start with an initial plenary session, where we will analyze and discuss a number of relevant sources on the role of machines in geometry. In fact, in the first half of the 18th century, some scholars worked on the practical realization of machines tracing transcendental curves by the physical solution of the inverse tangent problem. While the direct tangent problem (to find the tangent to a given curve) dates to ancient Greek geometry, the inverse tangent problem (to find the curve whose tangent satisfies certain conditions) is much more recent. Its first documented appearances date back to the 17th century: in a letter to Descartes, Florimond De Beaune asked for the curve having a constant subtangent or, in modern terms, the graph of an exponential function. The first actual construction of a curve under tangent conditions was displayed by the Parisian architect Claude Perrault in 1676, during a meeting that Leibniz also attended. Perrault considered the curve traced by a watch whose chain is dragged in a plane with friction. Because of the friction acting on the watch in the plane, the chain “in traction” is always tangent to the traced curve. As we will explore by reading original texts from Huygens, Leibniz, and others, the clock describes a tractrix. In the 18th century, the tractrix and the exponential constituted a benchmark to realize actual and precise devices to trace curves by inverse tangent conditions. The technical change behind such implementations is based on the passage from the use of a dragged point to that of a wheel that imposes the direction. However, even though this change was not commented on by any author at the time, it is not a mere technical improvement. Indeed, the dragged-point principle is not directly linked to the tangent but to the minimization of the displacement (that corresponds to the tangent in some elementary cases); on the other side, the wheel directly implements the definition of the tangent as the limit of the secants.
In the second part of the workshop, we will concretize the ideas of the previous discussion with the presentation of a new actual machine to be used in engaging laboratory experiences related to calculus. Such an artifact will be explored both through concrete manipulation and a digital simulation.
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Dopo la generale accettazione da parte dei matematici de La Géométrie di Descartes, nel XVII sec.... more Dopo la generale accettazione da parte dei matematici de La Géométrie di Descartes, nel XVII sec. la concezione e talvolta la realizzazione concreta di macchine per tracciare curve continua il suo ruolo fondazionale per le curve al di là dei confini cartesiani. Infatti, sebbene Descartes avesse proposto un metodo generale per giustificare l’introduzione delle curve grazie a delle macchine ideali, la classe di queste costruzioni è ben delimitata analiticamente ai problemi che possono essere risolti con l’algebra.
La possibilità di allargare la “legittimità” a curve trascendenti è uno degli obiettivi di Leibniz, che accusa Descartes di aver relegato ad un ruolo di inferiorità curve che non solo sono facilmente concepibili e realizzabili, ma che sono estremamente importanti per il problem solving matematico. Da un certo punto di vista Leibniz accetta il canone cartesiano per quanto riguarda il fatto che le macchine ideali per generare curve debbano essere “uno tractu” (in maniera moderna: con un grado di libertà), dall’altro non accetta i limiti che Descartes impone alle costruzioni. In particolare in questo intervento approfondiremo il ruolo ambivalente che i fili possono assumere nelle macchine ideali.
Infatti i fili si prestano a due utilizzi: possono riportare una lunghezza tra due punti, o essere utilizzati per rettificare perimetri di figure. A differenza di Descartes, che ammette come valido solo il primo metodo, Leibniz esplora le costruzioni che implicano l’uso “rettificatore” del filo, e propone interessanti modi per generare curve non algebriche. In alcuni esempi, mancando la descrizione delle sue macchine, abbiamo proposto delle possibili interpretazioni.
Concettualmente la geometria di Descartes si può considerare come un’estensione della geometria euclidea basata su equidistanza (come il compasso) e allineamento (come la riga). Il filo può essere visto come una riunificazione di entrambi (come per i “tenditori di funi” egizi), ed anzi, grazie alla modalità “rettificazione”, come una loro estensione. Negli anni 1673 -1675 Leibniz persegue questa strada, ma presto l'abbandona in quanto non riesce a trovarne un’opportuna controparte analitica. Per tale motivo successivamente perviene ad un metodo più generale, la risoluzione del problema inverso della tangente, analiticamente traducibile con l’introduzione delle derivate (movimento trazionale).
Se il XVII sec. vede i matematici atti ad ideare macchine teoriche per trovare le curve che risolvono determinati problemi, una volta abituati alla presenza di tali curve il bisogno del riferimento a macchine svanisce velocemente. Nel XVIII sec. sono però da citare gli interventi di alcuni italiani nella creazione di macchine per tracciare curve trascendenti, macchine che, a differenza del XVII sec., non rivestono più tanto un ruolo teorico e fondazionale, ma soprattutto pratico e didattico, come visibile nelle macchine trazionali del Gabinetto di Filosofia di Giovanni Poleni a Padova. Un approccio didattico che ha tutt’oggi qualcosa da insegnarci.
Bibliografia
V. Blasjo, Transcendental Curves in the Leibnizian Calculus, Academic Press, 2017.
H. J. M. Bos, Redefining Geometrical Exactness, Descartes' Transformation of the Early Modern Concept of Construction, Springer-Verlag, New York, 2001.
D. Tournès, La construction tractionnelle des équations différentielles, Blanchard, Paris, 2009.
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Books by Davide Crippa
The algebrization of mathematics during the 17th and 18th centuries , 2023
This book explores the major historical phenomenon of the algebraization of mathematics in the se... more This book explores the major historical phenomenon of the algebraization of mathematics in the second half of the 17th and 18th centuries, offering a
broader understanding of the consolidation of analytic geometry and infinitesimal calculus as disciplines. The authors examine the external (intellectual, geographical, and political) factors that influenced these transformations and shed light on the process of acquisition and integration of analytical mathematics into traditional curricula. Drawing on new trends in historiography of science, this book emphasizes the importance of "dwarfs", that is mathematicians but also technicians, artisans, military personnel, engineers, and architects, often ignored or marginalized in traditional histories, in the circulation of original mathematical knowledge, and of peripheral countries such as Italy and Spain as important sites for the appropriation and production of such knowledge.
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Book Reviews by Davide Crippa
Metascience, 2021
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Thesis Chapters by Davide Crippa
Papers by Davide Crippa
In this paper, we shall survey the content and structure of the course in Elementary Mathematics at the Charles-Ferdinand University in Prague. The teaching of mathematics at this college, which is fairly well documented, represents a fruitful case study for assessing the circulation of modern ideas in mathematical teaching. These ideas were promoted by contemporary authors like Christian Wolff or Nicolas Louis de la Caille, and were assimilated by the local scholarly community.
Since the present study represents no more than a preliminary exploration, our survey will cover only the period running from circa 1750 up to the year 1784. The terminus a quo here marks the publication of the first compendia written by a Jesuit professor from Prague and the terminus ad quem marks the end of Latin as the language of higher education. The switch to German as the official language of teaching introduced important changes, such as the adoption of new German textbooks and the end of Jesuit cultural hegemony even in scientific writings.
arcs. In this way, the previously mentioned curves resulted geometrical insofar as they could be generated by one and continuous motion.
by redefining the operation of multiplication in order to include both operations among segments and among numbers. Finally, we question about the significance of Descartes’ algebraic thought. Although the goal of Descartes’ Géométrie is to solve geometric problems, his first readers emphasized the role of algebra as a study of relations.
Drafts by Davide Crippa
Conference Presentations by Davide Crippa
The workshop will start with an initial plenary session, where we will analyze and discuss a number of relevant sources on the role of machines in geometry. In fact, in the first half of the 18th century, some scholars worked on the practical realization of machines tracing transcendental curves by the physical solution of the inverse tangent problem. While the direct tangent problem (to find the tangent to a given curve) dates to ancient Greek geometry, the inverse tangent problem (to find the curve whose tangent satisfies certain conditions) is much more recent. Its first documented appearances date back to the 17th century: in a letter to Descartes, Florimond De Beaune asked for the curve having a constant subtangent or, in modern terms, the graph of an exponential function. The first actual construction of a curve under tangent conditions was displayed by the Parisian architect Claude Perrault in 1676, during a meeting that Leibniz also attended. Perrault considered the curve traced by a watch whose chain is dragged in a plane with friction. Because of the friction acting on the watch in the plane, the chain “in traction” is always tangent to the traced curve. As we will explore by reading original texts from Huygens, Leibniz, and others, the clock describes a tractrix. In the 18th century, the tractrix and the exponential constituted a benchmark to realize actual and precise devices to trace curves by inverse tangent conditions. The technical change behind such implementations is based on the passage from the use of a dragged point to that of a wheel that imposes the direction. However, even though this change was not commented on by any author at the time, it is not a mere technical improvement. Indeed, the dragged-point principle is not directly linked to the tangent but to the minimization of the displacement (that corresponds to the tangent in some elementary cases); on the other side, the wheel directly implements the definition of the tangent as the limit of the secants.
In the second part of the workshop, we will concretize the ideas of the previous discussion with the presentation of a new actual machine to be used in engaging laboratory experiences related to calculus. Such an artifact will be explored both through concrete manipulation and a digital simulation.
La possibilità di allargare la “legittimità” a curve trascendenti è uno degli obiettivi di Leibniz, che accusa Descartes di aver relegato ad un ruolo di inferiorità curve che non solo sono facilmente concepibili e realizzabili, ma che sono estremamente importanti per il problem solving matematico. Da un certo punto di vista Leibniz accetta il canone cartesiano per quanto riguarda il fatto che le macchine ideali per generare curve debbano essere “uno tractu” (in maniera moderna: con un grado di libertà), dall’altro non accetta i limiti che Descartes impone alle costruzioni. In particolare in questo intervento approfondiremo il ruolo ambivalente che i fili possono assumere nelle macchine ideali.
Infatti i fili si prestano a due utilizzi: possono riportare una lunghezza tra due punti, o essere utilizzati per rettificare perimetri di figure. A differenza di Descartes, che ammette come valido solo il primo metodo, Leibniz esplora le costruzioni che implicano l’uso “rettificatore” del filo, e propone interessanti modi per generare curve non algebriche. In alcuni esempi, mancando la descrizione delle sue macchine, abbiamo proposto delle possibili interpretazioni.
Concettualmente la geometria di Descartes si può considerare come un’estensione della geometria euclidea basata su equidistanza (come il compasso) e allineamento (come la riga). Il filo può essere visto come una riunificazione di entrambi (come per i “tenditori di funi” egizi), ed anzi, grazie alla modalità “rettificazione”, come una loro estensione. Negli anni 1673 -1675 Leibniz persegue questa strada, ma presto l'abbandona in quanto non riesce a trovarne un’opportuna controparte analitica. Per tale motivo successivamente perviene ad un metodo più generale, la risoluzione del problema inverso della tangente, analiticamente traducibile con l’introduzione delle derivate (movimento trazionale).
Se il XVII sec. vede i matematici atti ad ideare macchine teoriche per trovare le curve che risolvono determinati problemi, una volta abituati alla presenza di tali curve il bisogno del riferimento a macchine svanisce velocemente. Nel XVIII sec. sono però da citare gli interventi di alcuni italiani nella creazione di macchine per tracciare curve trascendenti, macchine che, a differenza del XVII sec., non rivestono più tanto un ruolo teorico e fondazionale, ma soprattutto pratico e didattico, come visibile nelle macchine trazionali del Gabinetto di Filosofia di Giovanni Poleni a Padova. Un approccio didattico che ha tutt’oggi qualcosa da insegnarci.
Bibliografia
V. Blasjo, Transcendental Curves in the Leibnizian Calculus, Academic Press, 2017.
H. J. M. Bos, Redefining Geometrical Exactness, Descartes' Transformation of the Early Modern Concept of Construction, Springer-Verlag, New York, 2001.
D. Tournès, La construction tractionnelle des équations différentielles, Blanchard, Paris, 2009.
Books by Davide Crippa
broader understanding of the consolidation of analytic geometry and infinitesimal calculus as disciplines. The authors examine the external (intellectual, geographical, and political) factors that influenced these transformations and shed light on the process of acquisition and integration of analytical mathematics into traditional curricula. Drawing on new trends in historiography of science, this book emphasizes the importance of "dwarfs", that is mathematicians but also technicians, artisans, military personnel, engineers, and architects, often ignored or marginalized in traditional histories, in the circulation of original mathematical knowledge, and of peripheral countries such as Italy and Spain as important sites for the appropriation and production of such knowledge.
Book Reviews by Davide Crippa
In this paper, we shall survey the content and structure of the course in Elementary Mathematics at the Charles-Ferdinand University in Prague. The teaching of mathematics at this college, which is fairly well documented, represents a fruitful case study for assessing the circulation of modern ideas in mathematical teaching. These ideas were promoted by contemporary authors like Christian Wolff or Nicolas Louis de la Caille, and were assimilated by the local scholarly community.
Since the present study represents no more than a preliminary exploration, our survey will cover only the period running from circa 1750 up to the year 1784. The terminus a quo here marks the publication of the first compendia written by a Jesuit professor from Prague and the terminus ad quem marks the end of Latin as the language of higher education. The switch to German as the official language of teaching introduced important changes, such as the adoption of new German textbooks and the end of Jesuit cultural hegemony even in scientific writings.
arcs. In this way, the previously mentioned curves resulted geometrical insofar as they could be generated by one and continuous motion.
by redefining the operation of multiplication in order to include both operations among segments and among numbers. Finally, we question about the significance of Descartes’ algebraic thought. Although the goal of Descartes’ Géométrie is to solve geometric problems, his first readers emphasized the role of algebra as a study of relations.
The workshop will start with an initial plenary session, where we will analyze and discuss a number of relevant sources on the role of machines in geometry. In fact, in the first half of the 18th century, some scholars worked on the practical realization of machines tracing transcendental curves by the physical solution of the inverse tangent problem. While the direct tangent problem (to find the tangent to a given curve) dates to ancient Greek geometry, the inverse tangent problem (to find the curve whose tangent satisfies certain conditions) is much more recent. Its first documented appearances date back to the 17th century: in a letter to Descartes, Florimond De Beaune asked for the curve having a constant subtangent or, in modern terms, the graph of an exponential function. The first actual construction of a curve under tangent conditions was displayed by the Parisian architect Claude Perrault in 1676, during a meeting that Leibniz also attended. Perrault considered the curve traced by a watch whose chain is dragged in a plane with friction. Because of the friction acting on the watch in the plane, the chain “in traction” is always tangent to the traced curve. As we will explore by reading original texts from Huygens, Leibniz, and others, the clock describes a tractrix. In the 18th century, the tractrix and the exponential constituted a benchmark to realize actual and precise devices to trace curves by inverse tangent conditions. The technical change behind such implementations is based on the passage from the use of a dragged point to that of a wheel that imposes the direction. However, even though this change was not commented on by any author at the time, it is not a mere technical improvement. Indeed, the dragged-point principle is not directly linked to the tangent but to the minimization of the displacement (that corresponds to the tangent in some elementary cases); on the other side, the wheel directly implements the definition of the tangent as the limit of the secants.
In the second part of the workshop, we will concretize the ideas of the previous discussion with the presentation of a new actual machine to be used in engaging laboratory experiences related to calculus. Such an artifact will be explored both through concrete manipulation and a digital simulation.
La possibilità di allargare la “legittimità” a curve trascendenti è uno degli obiettivi di Leibniz, che accusa Descartes di aver relegato ad un ruolo di inferiorità curve che non solo sono facilmente concepibili e realizzabili, ma che sono estremamente importanti per il problem solving matematico. Da un certo punto di vista Leibniz accetta il canone cartesiano per quanto riguarda il fatto che le macchine ideali per generare curve debbano essere “uno tractu” (in maniera moderna: con un grado di libertà), dall’altro non accetta i limiti che Descartes impone alle costruzioni. In particolare in questo intervento approfondiremo il ruolo ambivalente che i fili possono assumere nelle macchine ideali.
Infatti i fili si prestano a due utilizzi: possono riportare una lunghezza tra due punti, o essere utilizzati per rettificare perimetri di figure. A differenza di Descartes, che ammette come valido solo il primo metodo, Leibniz esplora le costruzioni che implicano l’uso “rettificatore” del filo, e propone interessanti modi per generare curve non algebriche. In alcuni esempi, mancando la descrizione delle sue macchine, abbiamo proposto delle possibili interpretazioni.
Concettualmente la geometria di Descartes si può considerare come un’estensione della geometria euclidea basata su equidistanza (come il compasso) e allineamento (come la riga). Il filo può essere visto come una riunificazione di entrambi (come per i “tenditori di funi” egizi), ed anzi, grazie alla modalità “rettificazione”, come una loro estensione. Negli anni 1673 -1675 Leibniz persegue questa strada, ma presto l'abbandona in quanto non riesce a trovarne un’opportuna controparte analitica. Per tale motivo successivamente perviene ad un metodo più generale, la risoluzione del problema inverso della tangente, analiticamente traducibile con l’introduzione delle derivate (movimento trazionale).
Se il XVII sec. vede i matematici atti ad ideare macchine teoriche per trovare le curve che risolvono determinati problemi, una volta abituati alla presenza di tali curve il bisogno del riferimento a macchine svanisce velocemente. Nel XVIII sec. sono però da citare gli interventi di alcuni italiani nella creazione di macchine per tracciare curve trascendenti, macchine che, a differenza del XVII sec., non rivestono più tanto un ruolo teorico e fondazionale, ma soprattutto pratico e didattico, come visibile nelle macchine trazionali del Gabinetto di Filosofia di Giovanni Poleni a Padova. Un approccio didattico che ha tutt’oggi qualcosa da insegnarci.
Bibliografia
V. Blasjo, Transcendental Curves in the Leibnizian Calculus, Academic Press, 2017.
H. J. M. Bos, Redefining Geometrical Exactness, Descartes' Transformation of the Early Modern Concept of Construction, Springer-Verlag, New York, 2001.
D. Tournès, La construction tractionnelle des équations différentielles, Blanchard, Paris, 2009.
broader understanding of the consolidation of analytic geometry and infinitesimal calculus as disciplines. The authors examine the external (intellectual, geographical, and political) factors that influenced these transformations and shed light on the process of acquisition and integration of analytical mathematics into traditional curricula. Drawing on new trends in historiography of science, this book emphasizes the importance of "dwarfs", that is mathematicians but also technicians, artisans, military personnel, engineers, and architects, often ignored or marginalized in traditional histories, in the circulation of original mathematical knowledge, and of peripheral countries such as Italy and Spain as important sites for the appropriation and production of such knowledge.