# ENUMERATION OF LINES OF THE THIRD ORDER

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Page 25 - Umbras, in his Enumeration of Lines of the Third Order, where he remarks (p. 25, ed. Talbot) : " And in the same manner as the circle, projecting its shadow, generates all the conic sections, so the five divergent parabolas, by their shadows, generate all other curves of the second genus. And thus some of the more simple curves of other genera might be found, which would form all curves of the same genus by the projection of their shadows on a plane.
Page 119 - The base and vertical angle of a triangle being given to find the locus of the intersection of the straight lines, drawn from the angles to the middle of the opposite sides.
Page 140 - O will give radii vectores of every magnitude from oo to 1, and parabolic arcs from a to 0 ; hence, while the numbers range from oo to 1 , the parabolic arcs range from oo to 0. When the number lies between 1 and 0, the radius vector representing it is drawn below the axis ; its extremity...
Page 140 - We may accordingly infer that the logarithm of any number is equal to the logarithm of its reciprocal, with the sign changed, since (sec0+tan0) (sec0— tan 0)=1.
Page 35 - In every algebraic equation, the coefficient of whose highest term is unity, the coefficient pi of the second term with its sign changed is equal to the sum of the roots. The coefficient...
Page 7 - ... or, which is the same thing, according to the number of points in which they can be cut by a straight line.
Page 25 - The Generation of Curves by Shadows. If the shadows of curves caused by a luminous point, be projected on an infinite plane, the shadows of conic sections will always be conic sections ; those of curves of the second genus will always be curves of the second genus; those of the third genus will always be curves of the third genus ; and so on ad infinitum. And in the same manner as the circle, projecting its shadow, generates all the conic sections, so the five divergent parabolas, by their shadows,...
Page 83 - Sur les courbes que Ton forme en coupant une surface courbe quelconque par un plan donne de position," has treated this subject in a somewhat different manner.
Page 34 - A convex or concave line is such that it cannot be cut by a straight line in more than two points ; the concavity of the intercepted portion is turned towards the straight line, and the convexity from it.
Page 140 - O, p, to F, may by its radii vectores represent all positive numbers from +00 to + 0, the two infinite branches of the parabola will be used in representing the logarithms of positive numbers from + oo to + 0 ; that is, the upper or positive branch of the parabola will be "used up...