232 
THE SQUARING OF THE CIRCLE 
by Nicomedes, some have supposed the ‘sister of the eochloid ’ 
(or conchoid) to be the quadratrix, but this seems highly im 
probable. There is, however, another possibility. Apollonius 
is known to have written a regular treatise on the Gochlias, 
which was the cylindrical helix. 1 It is conceivable that he 
might call the cochlias the ‘ sister of the eochloid ’ on the 
ground of the similarity of the names, if not of the curves. 
And, as a matter of fact, the drawing of a tangent to the 
helix enables the circular section of the cylinder to be squared. 
For, if a plane be drawn at right angles to the axis of the 
cylinder through the initial position of the moving radius 
which describes the helix, and if we project on this plane 
the portion of the tangent at any point of the helix intercepted 
between the point and the plane, the projection is equal to 
an arc of the circular section of the cylinder subtended by an 
angle at the centre equal to the angle through which the 
plane through the axis and the moving radius has turned 
from its original position. And this squaring by means of 
what we may call the ‘ subtangent ’ is sufficiently parallel to 
the use by Archimedes of the polar subtangent to the spiral 
for the same purpose to make the hypothesis attractive. 
Nothing whatever is known of Carpus’s curve ‘ of double 
motion ’. Tannery thought it was the cycloid; but there is no 
evidence for this. 
(8) Approximations to the value of n. 
As we have seen, Archimedes, by inscribing and cir 
cumscribing regular polygons of 96 sides, and calculating 
their perimeters respectively, obtained the approximation 
3y > 7t > 3yy (Measurement of a Circle, Prop. 3). But we 
now learn 2 that, in a work on Plinthides and Cylinders, he 
made a nearer approximation still. Unfortunately the figures 
as they stand in the Greek text áre incorrect, the lower limit 
being given as the ratio of /qacooe to y^vya, or 211875:67441 
i0 
{■= 3-141635), and the higher limit as the ratio of y fairy to 
yfirva or 197888 : 62351 {— 3-17377), so that the lower limit 
1 Pappus, viii, p. 1110. 20; Proclus on Eucl. I, p. 105. 5. 
2 Heron, Metrica, i. 26, p. 66. 13-17.