Touching the transcendentals: tractional motion from the birth of calculus to future perspectives

When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century but today, mainstream conception relegates geometry to be merely a tool of visualization. In this snapshot, however, we consider geometric and constructive components of calculus. We reinterpret \u201ctractional motion\u201d, a late 17th century method to draw transcendental curves, in order to reintroduce \u201cideal machines\u201d in math foundation for a constructive approach to calculus that avoids the concept of infinity

Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives

When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century but today, mainstream conception relegates geometry to be merely a tool of visualization. In this snapshot, however, we consider geometric and constructive components of calculus. We reinterpret “tractional motion”, a late 17th century method to draw transcendental curves, in order to reintroduce “ideal machines” in math foundation for a constructive approach to calculus that avoids the concept of infinity

Exploring a new geometric-mechanical artefact for Calculus

We introduce a geometric-mechanical artefact designed for laboratory activities related to Calculus topics (3D models and construction instructions are freely available online). With new capabilities and a new design, this instrument adopts some mechanisms historically introduced to solve inverse tangent problems (that analytically correspond to solving differential equations). By such an instrument, besides materially revealing the tangent to a curve (tangent mode), it is possible to trace the graph of exponential functions and parabolas starting from the geometrical properties of their tangent (curvigraph mode). Furthermore, one can perform transformations as derivatives and integrals (transformation mode). Our research project aims to study the use of this artefact mainly for secondary school students. In this paper, we present the analysis of its semiotic potential, referring to the instrumental approach and the Theory of Semiotic Mediation. We also focus on a secondary school teacher manipulating the artefact to identify exploration processes and gestures of usage. The analysis supports the choice of starting the exploration in the tangent mode and suggests that the artefact fosters the emergence of the idea of the tangent line

Tractional Motion Machines extend GPAC-generable functions

In late 17th century there appeared the Tractional Motion instruments, mechanical devices which plot the curves solving differential equations by the management of the tangent. In early 20th century Vannevar Bush\u2019s Differential Analyzer got the same aim: in this paper we\u2019ll compare the Differential Analyzer mathematical model (the Shannon\u2019s General Purpose Analog Computer, or GPAC) with the Tractional Motion Machine potentials. Even if we will not arrive in defining the class of the functions generated by Tractional Motion Machines, we\u2019ll see how this class will strictly extend the GPAC-generable functions