There has been a sad error of the geometers in their attempts to approximate the circumference of a circle, claimed John A. Parker in his thought-provoking book, Quadrature of the Circle (New York: Wiley & Sons, 1874 edition). A portion of Parker’s Proposition 1 hopefully can be seen above. By clicking on the image, you should be able to obtain a larger view of same.
Proposition 1 states as follows: “The circumference of a circle is greater than any number whatever of mean proportionals from an inscribed straight line.” Plate 1 which accompanies the proposition offers an “ocular demonstration” that the circumference must always be greater than any inscribed lines. Any and all “mean proportionals must forever continue to be inscribed straight lines, they can therefore never equal the circumference.”
The very fact that they are inscribed means the lines can never equal the circumference, claimed Parker.
(But suppose a calculus situation, where the number of inscribed lines is considered to keep increasing, up to infinity: Could the inscribed lines eventually equal the circumference in such a situation?)
This all relates to the “squaring the circle” problem. Squaring the circle is “the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.” But that is only the classical problem’s imposed limitation. Parker begins, in Proposition 1 (above), to rule out the straightedge limitation in favor of a spatial solution. (Background: “Squaring The Circle”, Ersjdamoo’s Blog entry of August 10, 2013.)
The straightedge limitation had been the sad error of the geometers in their attempts to square the circle. An additional difficulty proposed by Parker was that the circumference is the boundary of the circle; the circle itself does not contain the circumference. “Circumference” means a line circumscribing the figure. But geometers, according to Parker, have made the mistake of making the circumference coincide with the greatest diameter of the figure.
Extrapolating from John A. Parker’s findings, one J. Ralston Skinner used a key ratio found by Parker (6561: 20612) to arrive at the value of the ancient cubit measure. (Background: “Finding The Cubit”, Ersjdamoo’s Blog entry of August 12, 2013.)
In Skinner’s book, Key to the Hebrew-Egyptian Mystery in the Source of Measures (1894. Available from kessinger.net), the author began to see all sorts of heretofore curious things emerging into the light of day due to the path-breaking efforts of John A. Parker. For example, the so-called “mound builders” seem to have been obsessed with circles and squares in their weird constructions. Do these monumental circles and squares of the “mound builders” pertain somehow to the “squaring the circle” problem?