Search results for "tractional motion"

Tractional Motion Machines extend GPAC-generable functions

2012

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’s Differential Analyzer got the same aim: in this paper we’ll compare the Differential Analyzer mathematical model (the Shannon’s 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’ll see how this class will strictly extend the GPAC-generable functions.

Geometrical-mechanical artefacts for managing tangent concept

2012

A CONSTRUCTIVE APPROACH TO THE INFINITESIMAL ANALYSIS: EPISTEMOLOGIC POTENTIALS AND LIMITS OF THE "TRACTIONAL MOTION"

2012

Une quête d'exactitude : machines, algèbre et géométrie pour la construction traditionnelle des équations différentielles

2015

In La Géométrie, Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one—geometry became a mean of visualization, no longer of construction. Descartes’s…

Tractional Motion Machines: Tangent-Managing Planar Mechanisms as Analog Computers and Educational Artifacts

2012

Concrete and virtual machines play a central role in the both Unconventional Computing (machines as computers) and in Math Education (influence of artifacts on reaching/producing abstract thought). Here we will examine some fallouts in these fields for the Tractional Motion Machines, planar mechanisms based on some devices used to plot the solutions of differential equations by the management of the tangent since the late 17th century.

Geometrical-mechanical artefacts mediating tangent meaning: the Tangentograph

2012

This work deals with the didactical use of geometrical-mechanical artefacts to acquire tangent concept in vygotskian perspective. We adopt Rabardel’s theory on instru-mental approach to distinguish artefacts and instruments specially to evince the history-to-education ontogenesis-phylogenesis process. From this point of view we trace a historical-epistemological pathway for the tangent up to set an ad hoc didactical counterpart. Specifically in this paper we deepen the kinematical properties of the tangent (introducing the XVII century so called tractional motion) designing a laboratorial didactic pathway for 12th grade students with the use of a particular geometrical-mechanical artefact f…

A geometrical constructive approach to infinitesimal analysis: epistemological potential and boundaries of tractional motion

2014

Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geome…