Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . M : 0 K (which are any two radiiwhose values are known) by the angle between these radii,expressed in circular measure; the quotient will be the tangentof the constant angle of obliquity of the spiral. 217. Among the more interesting properties of this curveare the following: — Its involute is an equal logarithmic a light placed at the pole, the caustic — whether byreflection or refraction—would , be a logarithmic spiral. The discove
Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . M : 0 K (which are any two radiiwhose values are known) by the angle between these radii,expressed in circular measure; the quotient will be the tangentof the constant angle of obliquity of the spiral. 217. Among the more interesting properties of this curveare the following: — Its involute is an equal logarithmic a light placed at the pole, the caustic — whether byreflection or refraction—would , be a logarithmic spiral. The discovery of these properties of recurrence led JamesBernouilli to direct that this spiral be engraved on his tomb,with the inscription — Eadem Mutata Resurgo, which, freely trans-lated, is — / shall arise the same, though changed. Kepler discovered that the orbits of the planets and comets were conic sections having a focusat the centre of the sun. Newton proved that they would have described logarithmic spirals asthey travelled out into space had the attraction of gravitation been inversely as the cube instead ofthe square of the THE HYPEBBOLIC OE RECIPROCAL SPIRAL. 218. In this spiral the length of a radius vector is in inverse ratio to the angle through whichit turns. Like the logarithmic spiral it has an infinite number of convolutions aljout the pole, which itnever reaches. The invention of this curve is attributed to James Bernouilli, who showed that Newtons con-clusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, theinitial velocity of projection determining which trajectory was -described. iss. To oljtain points of the curve divide a circle m 5 8 (Fig. 125)into any number of equal parts, and on some initial radius 0 mlay off some unit, as an inch; on the second radius 0 2 take 0 7h 0 01/ —^; on the third —^, etc. For one-half the angle 0 the radius vector would evidently be 2 0 n, giving a point s outsi
Size: 1404px × 1779px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1890, booksubjec, booksubjectlettering
