This treatise gives a broad outline of the history of the problem of Apollonius, from the publication of the Pappi Alexandrini Mathematicae Collectiones by Federicus Commandinus (1588) to the last original elaboration by Edward Kasner (1903). This is a history of mathematical reasoning, but also of drawing and how it influenced thought. Drawing, in fact, allows geometry scholars to carry out two basic operations: to verify hypotheses and to analyze unknown properties suggested by scientific experience. In this sense, drawing, for the geometrician, is exactly what an experiment is for the physicist. Representation, therefore is experimentation. Recently the spread of digital representation has profoundly changed the characteristics of experimental drawing; more precisely it has improved accuracy and enabled layout in space. This change, together with a broader vision of methods, has today led to the review of solutions and procedures which in the past were limited by the sole use of a ruler and compass and by complications arising out of the necessity to project space onto a plane. A good example of the new way of developing a geometrical description is the Apollonian problem which today, as demonstrated in the treatise, can be solved, on a plane and in space, with a single and generalized procedure devised by Adriaan Van Roomen (1596). Returning to the comparison with physics, we can say that the still immature methods and the uncertain graphic techniques of scholars who over the years have tackled the Apollonian problem have had the same effect as Galileo's telescope: a limitation, indeed a serious limitation, to analysis and comprehension of the problem. In this same volume another two brief treatises are dedicated to Gaultier and Kasner. But, once the experimental value of geometrical drawing has been ascertained, also through these studies on past works, it is necessary to draw the proper conclusions with regard to the present descriptive geometry, with methods and techniques now enhanced by digital representation, is at last in a position to resume its scientific development, first by recovering possession of wide-ranging literature, which has been neglected far years, then to propose new and stronger solutions to the problems of the description of forms and new studies on their geometric properties.
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