SPIRAL, in geometry, a curve line of the circular kind, which, in its progress, recedes from its centre.
A spiral, according to Archimedes, its inventor, is thus generated : if a right line, as AB, (Plate Miscel. XIV. fig. 9.) having one end fixed at B, be equally moved round, so as with the other end A to describe the periphery of a circle ; and, at the same time, a point be con ceived to move forward equally from B towards A, in the right line BA, so as that the point describes that line, while the line generates the circle : then will the point, with its.two motions, describe the curve line B 1, 2, 3, 4, 5, &c. which is called the helix or spiral line ; and the plane space contained between the spi ral line and the right line BA is called the spiral space.
If also you conceive the point B to move twice as slow as the line A B, so as that it shall get but half way along the line B A, when that line shall have fermtid the circle ; and if then you imagine a new revolution to be made of the line carrying the point, so that they shall end their mo tion at last together, there will be formed a double spiral line, and the two spiral spaces, as you sec in the figure. From
the genesis of this curve, the following corollaries may be easily drawn. 1. The lines B 12, B 11, B 10, &c. making equal angles with the first and second spiral (as all B 12, B 10, B 8, &c.) are in arithmetical proportion. 2. The lines B 7, B 10, &c. drawn any how to the first spiral, are to one another as the arches of the circle intercepted between B A and those lines. 3. Any lines drawn from B to the second spiral, as B 18, B 22, &c. are to each other as the aforesaid arches, together with the whole periphery added on both sides. 4. The first spiral space is to the first circle as one to 3. And 5. The first spiral line is equal to half the periphery of the first circle ; for the radii of the sectors, and consequently the arch es,are in a simple arithmetical progression, while the periphery of the circle contains as many arches equal to the greatest ; wherefore the periphery to all those arch es is to the spiral lines as 2 to 1.