《100 Translation of Eulers E083Greve and Osler July 2 2016》.pdf

1 ON SOME PROPERTIES OF CONIC SECTIONS THAT ARE SHARED WITH INFINITELY MANY OTHER CURVED LINES. L. EULER Date: 1745. Section 1 1.Conic sections have several properties that are shared only among themselves; but they also have several properties in common with infinitely many other curves. For instance, the axis that cuts the plane in two, with the origin placed at the vertex of the curve, is shared with infinitely many curves, both algebraic and transcendental. This is obvious to anyone who looks at the nature of curves. But geometry has shown other properties, which at first glance seem to be unique to conic sections, that are also shared with other curves. It is evident that the properties, by which conic sections are defined, are really their own, and that they cannot be shared with any other curve. However we encounter other properties beyond these, some of which are not easily seen to be unique, or not, to conic sections. To remedy this confusion, we must analytically find all curves that may have a certain property, and if we find that conic sections are the only curves satisfying that property, we will be certain that the property is unique to conic sections. Geometry has already given solutions to several questions of nature here and there; solutions that have considerably extended the art of mathematical analysis. We propose, thus, to add here some other similar questions, taken from the idea of oblique-angled diameters, that are shared principally with conic sections. 2 Section 2 2. We consider the following property of parabolas: any straight line parallel to the major axis is an oblique-angled diameter that divides all parallel lines drawn under a certain angle inside of the parabola in two equal parts. Indeed, if AMFm is a parabola with major axis AD , and if we dra