Treated as a secondary concern by Euclid, Archimedes and Apollonius, the problem of tangents became far more pressing in the 17th century, when the idea of "approximating curves with straight lines" guided the research of Fermat, Torricelli, Roberval and geometers in general. This quest led them to our modern notions of the tangent, and hence the derivative, and of the asymptote, and hence the limit at infinity, laying the foundations of analysis.
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Tangents: a mathematical adventure
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Tangente: the mathematical adventure. Our magazine bears this name because it evokes one of those adventures that mathematics does so well: travelling through history, appearing, disappearing and then reappearing in another form, thanks to a new approach, before emerging as a more modern, more general—in short, more fully developed—notion.
The notion of a tangent was not a major concern for the geometers of antiquity, but some discussed it in their own way. In his Éléments (The Elements), Euclid (3rd century BCE) discusses the tangent to a circle, without mentioning other curves, and offers this definition in Book III: "A straight line which, meeting a circle and being produced, does not cut the circle is said to touch the circle." He even gives us a characteristic property of this famous line: "The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and no other straight line can be inserted into the space between the straight line and the circumference; moreover, the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle. " The "remaining" angle—the famous horn angle between the curve and its tangent—prompted much debate among geometers, who wondered, among other things, whether it could be bisected.
