ARCHYTAS 
247 
construction in three dimensions, determining a certain point 
as the intersection of three surfaces of revolution, (1) a right 
cone, (2) a cylinder, (3) a tore or anchor-ring with inner 
diameter nil. The intersection of the two latter surfaces 
gives (says Archytas) a certain curve (which is in fact a curve 
o' 
of double curvature), and the point required is found as the 
point in which the cone meets this curve. 
Suppose that AC, AB are the two straight lines between 
which two mean proportionals are to be found, and let A C be 
made the diameter of a circle and AB a chord in it. 
Draw a semicircle with AG as diameter, but in a plane at 
right angles to the plane of the circle ABC, and imagine this 
semicircle to revolve about a straight line through A per 
pendicular to the plane of ABC (thus describing half a tore 
with inner diameter nil). 
Next draw a right half-cylinder on the semicircle ABC as 
base; this will cut the surface of the half-tore in a certain 
curve. 
Lastly let CD, the tangent to the circle ABC at the point C, 
meet AB produced in D; and suppose the triangle ADC to 
revolve about AG as axis. This will generate the surface 
of a right circular cone; the point B will describe a semicircle 
BQE at right angles to the plane of ABC and having its 
diameter BE at right angles to AG; and the surface of the 
cone will meet in some point P the curve which is the inter 
section of the half-cylinder and the half-£ore.