Author(s):

Eduardo Veloso and Rita Bastos

The kinds of examples that are included in this article have been used for several years to introduce the history of geometry to prospective mathematics teachers and in workshops or courses for inservice teacher training. We believe modeling historical constructions through dynamic geometry greatly enhances the understanding of some fundamental mathematical ideas and provides insights of concepts that inspired geometers of the past.

In this report we discuss just four examples, but it is not difficult to find many other episodes in the history of geometry ready to be used with a similar didactical approach. But these four examples are rich enough to show, so we think, that *The Geometer’s Sketchpad* (*GSP*) models allows us to follow much more closely the true dynamic ideas of these geometers than the usual static illustrations, and at the same time will encourage investigation and comparison of different solutions for the same problems.

For example, take the method of double projection proposed by Albrecht Dürer (1471-1528) for the tracing of conic sections (section II of the report). When we follow the instructions with dynamic geometry software like Geometer’s Sketch Pad, *GSP*, we feel that the basic and ingenious ideas behind those instructions is the construction of a moving horizontal plane that will intersect the cone’s surface in a moving and changing circle whose horizontal projection enables us to find two moving points of the horizontal projection of the conic section. Without our modern technology, Dürer is constrained to substitute (say) eleven positions for the moving plane, obtaining 22 points of the conic section, and then using these points to trace the ellipse point by point. But our modern technology allows us to realize the basic idea: to construct the ellipse as the locus defined by two moving points. Also, if we have just an illustration, we are only able to see a static plane cutting the conic surface and a fixed conic section, for instance an ellipse. But with the help of the dynamic geometry software, we may define a moving intersecting plane and as a consequence a changing conic section (going from an ellipse to a parabola and to a hyperbola). The reader may try to do this with *GSP* or at least to follow the constructions presented in the sections below. Similar observations may be made when we consider the other examples in this report.

In the first section of the report, we present a text from Piero della Francesca (1416-1492), proposition I.25 of *De Prospettiva Pingendi.* Piero gives instructions to construct the perspective image (in the painter’s canvas) of a square given in the horizontal plane. The solution of Piero is to define a map from a square (interior and border), representing the horizontal plane, onto a trapezium (interior and border) representing the perspective plane. As usual, the drawing of Piero that illustrates his explanations depicts a 2D situation, and we will find in the following text some animations made with *GSP* that will try to explain the 3D ideas behind the 2D illustration. But perhaps the main contribution of the dynamic geometry, in this case, is to show that the map defined by Piero, when extended to the whole plane, may be identified with a plane projective transformation. This is not surprising, because the origin of this transformation is a central projection, but the software enables us to “confirm” this conjecture and to easily draw the respective vanishing line.

In section III, the subject is the tracing of tangents to the cycloid and the solutions proposed are those by Gilles Persone de Roberval (1602-1675 ) and by René Descartes (1596-1650). One of the most interesting aspects is the fact that the construction of the cycloid proposed by Roberval, that is, to consider the cycloid as generated by a point subjected to two uniform motions, one circular and the other straight, is the most appropriate one to be used with dynamic software to trace the cycloid as a locus of a point that moves around a circular path that simultaneously moves in a straight path. Also, the consideration and construction of prolate and curlate cycloids is very straightforward with the help of *GSP*.

The subject of section IV is based on the work of Gaspard Monge (1746-1818) (extracted from his *Descriptive Geometry* of 1799), describing one method of finding the tangent planes to a given sphere containing a given line. As with the previous tasks, the work in descriptive geometry is greatly simplified by the software for dynamic geometry. But for anyone who is a newcomer to descriptive geometry – and this is the case for almost all prospective and inservice teachers in Portugal – it is not an easy task to follow Monge’s drawings in double projection. But with a tool that can represent each step of a construction in cavalier perspective – that is not difficult to build in *GSP* – we may follow step by step those drawings and visualize the situation very easily. The main ideas of Monge’s method to find the two tangent planes are the following:

• To construct two conic surfaces with vertices in two points of the given line *e* and touching the sphere (in two circles *c* and *c*’) ;

• To obtain the two points of tangency of the planes and the sphere, i.e. R and S;

• Finally defining the two planes: R and *e* and S and *e*.

The developing of this idea will be easier to follow with the help of the cavalier perspective, as you may imagine in the following final illustration, showing in the left side the cavalier perspective and on the right side the Monge drawing:

Author(s):

Eduardo Veloso and Rita Bastos

The kinds of examples that are included in this article have been used for several years to introduce the history of geometry to prospective mathematics teachers and in workshops or courses for inservice teacher training. We believe modeling historical constructions through dynamic geometry greatly enhances the understanding of some fundamental mathematical ideas and provides insights of concepts that inspired geometers of the past.

In this report we discuss just four examples, but it is not difficult to find many other episodes in the history of geometry ready to be used with a similar didactical approach. But these four examples are rich enough to show, so we think, that *The Geometer’s Sketchpad* (*GSP*) models allows us to follow much more closely the true dynamic ideas of these geometers than the usual static illustrations, and at the same time will encourage investigation and comparison of different solutions for the same problems.

For example, take the method of double projection proposed by Albrecht Dürer (1471-1528) for the tracing of conic sections (section II of the report). When we follow the instructions with dynamic geometry software like Geometer’s Sketch Pad, *GSP*, we feel that the basic and ingenious ideas behind those instructions is the construction of a moving horizontal plane that will intersect the cone’s surface in a moving and changing circle whose horizontal projection enables us to find two moving points of the horizontal projection of the conic section. Without our modern technology, Dürer is constrained to substitute (say) eleven positions for the moving plane, obtaining 22 points of the conic section, and then using these points to trace the ellipse point by point. But our modern technology allows us to realize the basic idea: to construct the ellipse as the locus defined by two moving points. Also, if we have just an illustration, we are only able to see a static plane cutting the conic surface and a fixed conic section, for instance an ellipse. But with the help of the dynamic geometry software, we may define a moving intersecting plane and as a consequence a changing conic section (going from an ellipse to a parabola and to a hyperbola). The reader may try to do this with *GSP* or at least to follow the constructions presented in the sections below. Similar observations may be made when we consider the other examples in this report.

In the first section of the report, we present a text from Piero della Francesca (1416-1492), proposition I.25 of *De Prospettiva Pingendi.* Piero gives instructions to construct the perspective image (in the painter’s canvas) of a square given in the horizontal plane. The solution of Piero is to define a map from a square (interior and border), representing the horizontal plane, onto a trapezium (interior and border) representing the perspective plane. As usual, the drawing of Piero that illustrates his explanations depicts a 2D situation, and we will find in the following text some animations made with *GSP* that will try to explain the 3D ideas behind the 2D illustration. But perhaps the main contribution of the dynamic geometry, in this case, is to show that the map defined by Piero, when extended to the whole plane, may be identified with a plane projective transformation. This is not surprising, because the origin of this transformation is a central projection, but the software enables us to “confirm” this conjecture and to easily draw the respective vanishing line.

In section III, the subject is the tracing of tangents to the cycloid and the solutions proposed are those by Gilles Persone de Roberval (1602-1675 ) and by René Descartes (1596-1650). One of the most interesting aspects is the fact that the construction of the cycloid proposed by Roberval, that is, to consider the cycloid as generated by a point subjected to two uniform motions, one circular and the other straight, is the most appropriate one to be used with dynamic software to trace the cycloid as a locus of a point that moves around a circular path that simultaneously moves in a straight path. Also, the consideration and construction of prolate and curlate cycloids is very straightforward with the help of *GSP*.

The subject of section IV is based on the work of Gaspard Monge (1746-1818) (extracted from his *Descriptive Geometry* of 1799), describing one method of finding the tangent planes to a given sphere containing a given line. As with the previous tasks, the work in descriptive geometry is greatly simplified by the software for dynamic geometry. But for anyone who is a newcomer to descriptive geometry – and this is the case for almost all prospective and inservice teachers in Portugal – it is not an easy task to follow Monge’s drawings in double projection. But with a tool that can represent each step of a construction in cavalier perspective – that is not difficult to build in *GSP* – we may follow step by step those drawings and visualize the situation very easily. The main ideas of Monge’s method to find the two tangent planes are the following:

• To construct two conic surfaces with vertices in two points of the given line *e* and touching the sphere (in two circles *c* and *c*’) ;

• To obtain the two points of tangency of the planes and the sphere, i.e. R and S;

• Finally defining the two planes: R and *e* and S and *e*.

The developing of this idea will be easier to follow with the help of the cavalier perspective, as you may imagine in the following final illustration, showing in the left side the cavalier perspective and on the right side the Monge drawing:

Author(s):

Eduardo Veloso and Rita Bastos

The descriptions of the four lessons are in two files, one in English and one in French. To access the English file, click here. To access the French file, click here. It will probably be best to download these files to your own computer to read them.

To see the dynamic versions of the constructions, you will need to have *Geometer's Sketchpad* installed on your computer. Then you can download the GSP files for the four constructions, either in English or in French:

English:

1. Piero

2. Dürer

4. Monge

French:

1. Piero

2. Dürer

4. Monge