About the Book
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...will be 2a ir, which is the measuring arc for the first revolution of PC; therefore, since PA or a is then the radius vector, a = a 2a n 1 whence a =. 2ir Substituting this value of a in (3) gives the equation of the spiral of Archimedes. When n is equal to one half, (1) becomes r = av or r2 = a2v, which is the equation of the parabolic spiral, being of the same form as that of the parabola; for substituting y for r, y/(2p) for a, and x for v gives y2 = 2px. (4) With 2p as radius draw the circle ABC, divide its 'circumference into any number of equal parts, as six, and draw through its center P, the divisional lines DV, EE', FF'. With-(2p + 1), -(2p + 2), i (2 + 3), and-(2p + 4), (1, 2, 3, and 4 being values given x as in the construction of the parabola) as radii, draw the arcs Aa, AbC, Ac, and Ad, having their centers in the line AG; then with Pa, Pb, Pc, and Pd, as radii, draw the arcs aa', bb', cc', and dd', and the curve drawn from P through a', c', d' will be the required spiral. In proof, it will be seen that AP: PC is equal to Pb2 or Pb'2 (see Euclid, proposition 35, Book III); hence, since AP = 2p and PC = x, if y be represented by Pb--Pb', then y2---2px. Also, when x--0, y = 0; therefore the spiral commences at P, its pole. When n is equal to--1, (1) becomes The curve represented by this equation is called the hyperbolic spiral on account of its analogy to that of the hyperbola when referred to its center and asymptote. With a as a radius draw the circle ABC and divide its circumference, 2a ir, into any number of equal parts, as six; then giving to v the Sai r 4a w values 2a ir, _ 3 3 air, etc., the corresponding 1 values of r will be, 2t 3 3 1----, --, etc. Let Pa, Pb, Pc, Pd, etc. represent these D 7T 4 7T TT values of r; then the curve drawn t...
