Talk:Method of normals
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A reference + Is it over-complicated?
[edit]I've seen a video that explained both methods with references; the first one is "R. Descartes, La Geometrie, Trans. David Eugene Smith and Marcia L. Latham. Open Court Publishing Company: La Salle. (1952), pp. 95–112."
The method is called rigorous in the video, but I don't entirely understand it because I don't understand the contemporary definition of tangent lines (a line such that no other straight line could fall between it and the curve). However, I've invented a simpler version that has a similar main part (i.e., the one the rigorousity of which I don't understand): the tangent line is the one for which the intersection corresponds to a double solution to the equation you can get by substituting one of them (defined parametrically) into the other (defined by an equation). My version uses straight lines instead of circles and can find tangent lines to the Folium of Decartes. You can find it in this comment. Most likely the definition for tangent lines somehow allows for making the point about circles, but not straight lines, but maybe humans are somehow so dumb in some aspects that they all overlooked this. Even in the latter case, I don't think my discovery is really significant, but it shows something psychological.