Because straight lines have been made the basis of area in mathematics, argued John A. Parker in his book, Quadrature of the Circle, “some compensation must be made to the circumference of the circle” because of the difference in relative property. There is a difference between the properties of straight lines and the properties of curved lines.
One example of this difference in relative properties between curved lines and straight lines was given by Parker in his Proposition 1 from the first chapter of his book: the circumference of a circle must always be greater than any inscribed lines. (Background: “Sad Error of the Geometers”, Ersjdamoo’s Blog entry of August 13, 2013)
Chapter one of Parker’s book (1874 edition) sought to prove there were fundamental errors in the analysis of geometers. There Parker had demonstrated “that there is an essential difference in the properties of straight lines and curved lines…” My sense of Parker’s chapter one is that it shows how the “squaring the circle” problem cannot be solved if a limit of compass and straightedge is imposed. Because this impossibility is nowadays generally conceded, I will (for now anyway) skip ahead to chapter two of Parker’s book, where he inquires into what is true respecting the properties of curved lines. (Further background: “Squaring The Circle”, Ersjdamoo’s Blog entry of August 10, 2013.)
Propositions in chapter two of Parker’s book also begin with “1.” To avoid confusion, I will preface Parker’s chapter two propositions with a “2.” So, for example, Parker’s chapter two first proposition is here labeled Proposition 2-1.
The first part of Parker’s Proposition 2-1 says that all shapes formed of straight lines and equal sides have their areas equal to half the circumference multiplied by the least radius which the shape contains. I tried this out with a square having sides equal to 3 and least radius equal to 1.5. Half the circumference of said square would be 6. Multiplying that by 1.5 returns an area of 9, which is correct.
The second part of Parker’s Proposition 2-1 says that the circle has its area equal to half the circumference multiplied by the radius. I tried this out with a circle having a radius equal to 1. The circumference of said circle (C= 2*PI*radius) was found by me to be 6.283185307. Half of this circumference is 3.141592654. Using Parker’s formula of half the circumference multiplied by the radius returns an area of 3.141592654. Checking this against the usual formula of A=Pi times radius squared returns the same answer as the Parker formula.
What I take to be the gist of what Parker is saying in his Proposition 2-1 is the subtle difference of least radius involved in determining area for shapes formed of straight lines and equal sides. For a circle, the radius is always the same. For straight-lined shapes the radius can vary.
Any intelligent comments and/or critiques regarding the above are welcomed.