THE SQUARING OF THE CIRCLE 
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curve, where it met the straight line AD. Unless indeed any 
one should assert that the curve is conceived to be produced 
further, in the same way as we suppose straight lines to be 
produced, as far as AD. But this does not follow from the 
assumptions made; the point G can only be found by first 
assuming (as known) the ratio of the circumference to the 
straight line.’ 
The second of these objections is undoubtedly sound. The 
point G can in fact only be found by applying the method 
of exhaustion in the orthodox Greek manner; e.g. we may 
first bisect the angle of the quadrant, then the half towards 
AD, then the half of that and so on, drawing each time 
from the points F in which the bisectors cut the quadratrix 
perpendiculars FH on AD and describing circles with AF 
as radius cutting AD in K. Then, if we continue this process 
long enough. IIK will get smaller and smaller and, as G lies 
between II and K, we can approximate to the position of G as 
nearly as we please. But this process is the equivalent of 
approximating to n, which is the very object of the whole 
construction. 
As regards objection (1) Hultsch has argued that it is not 
valid because, with our modern facilities for making instru 
ments of precision, there is no difficulty in making the two 
uniform motions take the same time. Thus an accurate clock 
will show the minute hand describing an exact quadrant in 
a definite time, and it is quite practicable now to contrive a 
uniform rectilinear motion taking exactly the same time. 
I suspect, however, that the rectilinear motion would be the 
result of converting some one or more circular motions into 
rectilinear motions; if so, they would involve the use of an 
approximate value of tt, in which case the solution would depend 
on the assumption of the very thing to be found. I am inclined, 
therefore, to think that both Sporus’s objections are valid. 
(/3) The Spiral of Archimedes. 
We are assured that Archimedes actually used the spiral 
for squaring the circle. He does in fact show how to rectify 
a circle by means of a polar subtangent to the spiral. The 
spiral is thus generated: suppose that a straight line with 
one extremity fixed starts from a fixed position (the initial