1. If a straight line drawn in a plane revolve at a uniform rate about one extremity which remains fixed and return to the position from which it started, and if, at the same time as the line revolves, a point move at a uniform rate along the straight line beginning from the extremity which remains fixed, the point will describe a spiral in the plane.
2. Let the extremity of the straight line which remains
fixed while the straight line revolves be called the origin* of the spiral.
3. And let the position of the line from which the straight line began to revolve be called the initial line* in the revolution .
4. Let the length which the point that moves along the straight line describes in one revolution be called the first distance, that which the same point describes in the second revolution the second distance, and similarly let the distances described in further revolutions be called after the number of the particular revolution.
5. Let the area bounded by the spiral described in the first revolution and the first distance be called the first area, that bounded by the spiral described in the second revolution and the second distance the second area, and similarly for the rest in order.
6. If from the origin of the spiral any straight line be drawn, let that side of it which is in the same direction as that of the revolution be called forward, and that which is in the other direction backward.
7. Let the circle drawn with the origin as centre and
the first distance as radius be called the first circle, that
drawn with the same centre and twice the radius the second circle,
and similarly for the succeeding circles.
*The literal translation would of course be the "beginning of the spiral" and "the beginning of the revolution" respectively. But the modern names will be more suitable for use later on, and are therefore employed here.
If any number of straight lines drawn from the origin to meet the spiral make equal angles with one another, the lines will be in arithmetical progression.
[The proof is obvious.]
If a. straight line touch the spiral, it will touch it in
one point only.
Let 0 be the origin of the spiral, and BC a tangent to it.
If possible, let BC touch the spiral in two points P, Q. Join OP, 0Q3 and bisect the angle POQ by the straight line OR meeting the spiral in R.
Then [Prop. 12] OR is an arithmetic mean between OP and OQ, or
But in any triangle POQ, if the bisector of the angle POQ meets PQ in K,
Therefore OK < OR, and it follows that some point on BC between P and Q lies within the spiral. Hence BC cuts the spiral; which is contrary to the hypothesis.
If 0 be the origin, and P, Q two points on the first turn of the spiral, and if OP, OQ produced meet the 'first circle' AKP'Q' in P', Q' respectively', OA being the initial line, then
If BC be the tangent at P, any point on the spiral, PC being the 'forward' part of BC, and if OP be joined, the angle OPC is obtuse while the angle OPB is acute.
I. If OA be the initial line, A the end of the first turn of the spiral, and if the tangent to the spiral at A be drawn, the straight line OB drawn from 0 perpendicular to OA will meet the said tangent in some point B, and OB will be equal to the circumference of the 'first circle.'
II. If A' be the end of the second turn, the perpendicular OB will meet the tangent at A' in some point B', and OB' will be equal to 2 (circumference of 'second circle').
III. Generally, if An be the end of the nth turn, and OB meet the tangent at An in Bn, then
where cn is the circumference of the 'nth circle.'
I. If P be any point on the first turn of the spiral and OT be drawn perpendicular to OP, OT will meet the tangent at P to the spiral in some point T; and, if the circle drawn with centre O and radius OP meet the initial line in K, then OT is equal to the arc of this circle between K and P measured in the 'forward' direction of the spiral.
II. Generally, if P be a point on the nth turn, and the notation be as before, while p represents the circumference of the circle with radius OP,
OT= (n 1)p + arc KP (measured 'forward').