A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOK IV 381
Hence / lies on a certain curve.
Therefore E, its projection
on the plane ABO, also lies on a curve.
In the particular case where the given ratio of EF to the
arc CD is equal to the ratio of BA to the arc CA, the locus of
E is a quadratrix.
[The surface described by the straight line LH is a plectoid.
The shape of it is perhaps best realized as a continuous spiral
staircase, i.e. a spiral staircase with infinitely small steps.
The quadratrix is thus produced as the orthogonal projection
of the curve in which the plectoid is intersected by a plane
through BC inclined at a given angle to the plane ABC. It is
not difficult to verify the result analytically.]
(2) The second method uses a right cylinder the base of which
is an Archimedean spiral.
Let ABC be a quadrant of a circle, as before, and EF, perpendicular
at F to BC, a straight
line of such length that EF is
to the arc DC as AB is to the
arc ADC.«
Let a point on AB move uniformly
from A to B while, in the
same time, AB itself revolves
uniformly about B from the position BA to the position BC.
The point thus describes the spiral AGB. If the spiral cuts
BD in G,
or BG :
BA:BG = (arc ADC) :
(arc DC) = BA :
(arc ADC).
(arc DC),
Therefore BG = EF.
Draw GK at right angles to the plane ABC and equal to BG.
Then GK, and therefore K, lies on a right cylinder with the
spiral as base.
But BK also lies on a conical surface with vertex B such that
its generators all make an angle of \tt with the plane ABC.
Consequently K lies on the intersection of two surfaces,
and therefore on a curve.
Through K draw LK1 parallel to BD, and let BL, EI be at
right angles to the plane ABC.
Then LKI, moving always parallel to the plane ABC, with
one extremity on BL and passing through K on a certain
382 PAPPUS OF ALEXANDRIA
curve, describes a certain plectoid, which therefore contains the
point /.
Also IE = EF, IF is perpendicular to BG, and hence IF, and
therefore 7, lies on a fixed plane through BG inclined to ABG
at an angle of ^w.
Therefore I, lying on the intersection of the plectoid and the
said plane, lies on a certain curve. So therefore does the
projection of I on ABG, i.e. the point E.
The locus of E is clearly the quadratrix.
[This result can also be verified analytically.]
(S) Digression: a spiral on a sphere.
Prop. 30 (chap. 35) is a digression on the subject of a certain
spiral described on a sphere, suggested by the discussion of
a spiral in a plane.
Take a hemisphere bounded by the great circle KLM,
with H as pole. Suppose that the quadrant of a great circle
HNK revolves uniformly about the radius HO so that K
describes the circle KLM and returns to its original position
at K, and suppose that a point moves uniformly at the same
time from H to
K at such speed that the point arrives at K
at the same time that HK resumes its original position. The
point will thus describe a spiral on the surface of the sphere
between the points H and K as shown in the figure.
Pappus then sets himself to prove that the portion of the
surface of the sphere cut off towards the pole between the
spiral and the arc HNK is to the surface of the hemisphere in