A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections

Hodges & Smith, 1852 - Curves, Algebraic - 316 pages
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Contents

 CHAPTER IV 195 CHAPTER V 205 CHAPTER VI 226 Geometrical Methods 243 Rectification of Curves 255
 Trajectories and Involutes 271 Successive Evolutes of Curves 279 On Elimination 285 On Imaginary Points and Lines 301 On the ultimate Involute of a Curve 1307 316

Popular passages

Page 5 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 122 - The locus of a point whose distances from two fixed points are in a constant ratio (not one of equality) is a circle.
Page 269 - Archimedes depend on the same formulse as in the case of the parabola.* The following is a method by which rectification may in general be reduced to quadratures, and by which, in particular, Van Huraet rectified the semicubical parabola, the first curve rectified. (Lardner's Algebraic Geometry, p. 464.) Produce each ordinate until the whole length be in a constant ratio to the corresponding normal divided by the old ordinate, and the locus of the extremity of the produced ordinate is a curve whose...
Page 206 - This curve is generated by the motion of a point on the circumference of a circle which rolls along a right fcline.
Page 225 - Cum autem ob proprietatem tam singularem tamque admirabilem mire mihi placeat Spira haec mirabilis, sic ut ejus contemplatione satiari vix queam; cogitavi, illam ad varias res symbolice repraesentandas non inconcinne adhiberi posse. Quoniam enim semper sibi similem...
Page 223 - ... 0, &c. The same right line then meets the curve in an infinity of points, and the curve is transcendental. Let us first take the spiral of Archimedes, which is the path described by a point receding uniformly from the origin, while the radius vector on which it travels...
Page 118 - ... refraction of a circle. Let a circle be described through A, the radiant point, and R, the point of incidence, to touch OR ; then the point B is given, since OA . OB = OR1.
Page 247 - Ex. 4. The three sides of a triangle pass through fixed points, and two...
Page 204 - This lemniscata is the locus of the foot of the perpendicular from the centre on the tangent to an equilateral hyperbola.
Page 225 - ... non inconcinne adhiberi posse. Quoniam enim semper sibi similem et eandem spiram gignit, utcunque volvatur...