- In the Mathematical Laboratory of the University Museum of Natural
History and Scientific Instruments there are about 160 mathematical machines
built by teachers of Liceo A.Tassoni in the scope of an innovation project
for the teaching of geometry developed by the Nucleo di Ricerca in Storia
e Didattica della Matematica (University of Modena) . The project , coordinated
by prof. Mariolina Bartolini Bussi (bartolini@unimo.it)
has produced mathematical machines, didactical itineraries, animation films
and simulations by computer. With these teaching
aids we introduce a historical dimension and a manipulative and visual
one in our classrooms. Our didactical research purpose is the historical
contextualization of problems, of theories and of methods.Many "machines"
realize projects or ideas of mathematicians, from the Ancient Greek up
to now. For the use of these "machines" in the classroom, students have
to elaborate abstract themes and proofs.
- Conics families.
- Complex numbers: intersections of a parabola with a line.
- Complex numbers: intersections of a cubic curve with a line.
- Pithagoras theorem and affinities.
- Carnot theorem and affinities.
- Shadows and affinities.

Classroom activities have been developed on curricular themes using machines and also some seminars have been organized with teachers of different schools . Two exibitions were organized , in Modena in 1992 and in Torino in 1996. We are collaborating with Vierkant Foundation -Amsterdam (http://www.cs.vu.nl./~vierkant/) and with the group Cabri-Geometre of Grenoble (http//www-cabri.imag.fr/).

The research project on Mathematical Machines had been financed by the
Municipality of Modena, by CNR, by MURST and by University of Modena.

**Mathematical machines of our laboratory can be divided in the following
groups:**

**1) Pantographs**,i.e. linkages that
allow to draw the image of a figure under a given transformation. Geometrical
transformations considered are: symmetry to an axis, simmetry to a point,
translation, rotation, glide reflection, homothety, similarity, affinity,
inversion, perspectivity. In some machines, two or more linkages are connected
: so that it is possible to visualize the result of the composition of
two transformations.

**2) Curve drawers**, i.e. machines (built
by wood, plexiglas, metal bars and stretched threads) which can force a
point or a straight line to move along a given trajectory. They are: machines
drawing a straight line (Kempe and Hart), Descartes machines, Cavalieri
machines and DeL'Hospital machines (drawing conics), ellipsographs (Van
Schooten), Newton square drawing a strophoid and a cissoid, mecanical linkages
for lemniscate, for conchoid, for limancon,machines drawing envelopes.

**3) Models **(built by wood, plexiglas,
metal bars and stretch threads) by which it is possible to show many important
properties and theorems; i.e. cones, to derive the symptom of conic sections,
cones and cylinders to show Dandelin theorems, models to show conics as
metamoorphoses of circles, Durer perspectograph, models to illustrate the
generation of geometrical transformations in the space.

**4) Machines** that were built to solve
very important problems in the history of mathematics, like mesolabon,
trisectors, the squares of Bombelli ecc.

The didactical research studies of N.R.S.D.M.have concerned particular themes like :the concept of function, complex numbers, probability, geometrical transformations, conics. (References)