[Fig. 34] The only difference between the working of this figure—the Hyperbola—and the Parabola is that the lines which in the Parabola were drawn parallel to G B, are here drawn to a point E on C D produced, C D being equal to X 3 ([Fig. 31]). This point E is found by drawing the line from 7 on D B to E on C D produced, where C E equals twice X O ([Fig. 31]).
[Fig. 35] To describe an Archimedean spiral of any number of revolutions—say three, the longest radius A B being given.
Fig. 35.—Archimedean Spiral.
Divide the radius A B into three equal parts for the three revolutions. With B as centre and B A as radius describe a circle, and divide it into any number of equal parts—say eight, by drawing four diameters. Each of the three divisions on A B is divided into eight equal parts. With centre B and the point of each succeeding division as radius, describe arcs, meeting in following order the next nearest diameter as shown at arcs 1 1´´, 2 2´´, 3 3´´, &c. Through point 8 with radius B 8, the second division, describe a circle, and through point 16 with centre B describe a circle. In these two divisions arcs are drawn as described above for the division A 8, &c., to the next nearest diameter. The spiral is then drawn through the points thus formed on the diameters, which mark its path as at 1´, 2´, 3´, &c., until it ends in its centre at B.