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Professor Henk Bos, Unversitaet Utrecht
During most of the early modern period the representation of geometrical objects by their equations was not accepted as sufficient answer to the question “which object do you mean?” Geometrical problems therefore, even when solved by means of algebra, were considered adequately solved only when the required object was somehow constructed. However, the new problems, especially those involving curves, were definitely beyond the reach of the standard Euclidean means of construction, circles and straight lines (“ruler and compass”). There was, then, an intense discussion on the question which constructions could be considered adequate for the solution of these new problems. Mathematicians described the requirements for constructions to be acceptable in terms as “exact”, “correct”, “properly geometrical” or just “geometrical”. I use the term “interpretation of exactness” (or “redefinition of exactness”) for the explicit discourses about the legitimacy of constructions as well as for the practices that arose in finding new means of construction where traditional ones were seen as inadequate. The discussions did not result in a commonly accepted practice, rather they fell into oblivion.
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