Although he is best known as a cofounder of the moon hoax movement, one of Ralph Rene's
crowning achievements was revealing an even bigger hoax perpetrated by the math and science
community for over two centuries, by verifying Dan W. Gaddy's "proof" that the traditional
value of pi that we all know and love is wrong.
According to Rene's interpretation of this line-and-circle construction, the absolute
value of pi should be equal to (the square root of two plus the square root of three).
Armed with this newfound knowledge, Rene wrote a 12-page pamphlet entitled, "On Squaring
the Circle," in which Rene offers his "arithmetic" proof that Gaddy's pi is the correct value,
but did he actually succeed in proving anything?
Well, before we follow Rene down this rabbit hole, let's review the ground rules that
Gaddy obeyed for his line-and-circle construction.
Around 300BC, Euclid of Alexandria gave us five postulates, three of which were useful to Gaddy.
Postulates are statements that we assume to be true, unlike theorems, which must be proven.
Postulate One: Any two points can be connected by a straight line segment
by using a straightedge or ruler.
Postulate Two: Any line segment can be extended by using that same straightedge or ruler.
Postulate Three: You can draw a circle with any given center and radius by using a compass.
A useful combination of postulates that Gaddy employs is constructing a perpendicular line
to a line segment at a given point.
You do this by drawing a circle around a reference point...
...extending the line segment, if necessary, to intersect the circle at two points, equidistant
from the reference point...
...drawing a couple of larger circles centered at these outside points...
...and then drawing a line through the points where these circles intersect.
This line is technically a perpendicular bisector.
But after chopping off the extension, it's simply a perpendicular line.
Another tool in Gaddy's bag is the Pythagorean Theorem.
When two line segments meet at a right angle...
...you can complete the triangle by constructing the hypotenuse.
The length of this hypotenuse is the square root of the sum of the squares
of the two legs.
Now, armed with these tools, and a few area formulas, we can breeze through Gaddy's construction...
Oh! And the most important "rule" for line-and-circle constructions is...
NO MEASUREMENTS ALLOWED!!!
We start with a point, which we'll call point-A...
...and, a second point, which we'll call point-B.
Did I say we were going to breeze through this?
We can connect these points with a line segment, which we'll call AB.
We then construct a circle, centered at point-A, using this line segment as a radius.
We can't measure the length of radius AB, but we can reference this length with a placeholder.
Let's call it "r" (for radius, of course).
Construct another radius perpendicular to segment AB.
The point where this radius intersects the circle is point-C.
The cord BC, connecting the two radii, is the hypotenuse of a right triangle, CAB.
According to the Pythagorean Theorem, the length of this cord is
the square root of two times r.
Draw another circle centered at point-C that goes through point-B.
The radius of this circle is the square root of two times r.
This circle has twice the area of the first.
Extend a radius from point-C, perpendicular to line segment AC.
This segment intersects the second circle at point-D.
The segment AD is the hypotenuse of another right triangle, ACD.
The length of segment AD is the square root of three times r.
Triangle ACD is not congruent to the previous triangle, CAB.
And, there are no definitions, theorems or postulates relating the areas of either triangle
to the beginning circle.
Draw another circle, centered at point-D, that passes through point-A.
The radius of this circle is the square root of three times r.
The area of this circle is three times the area of the first circle.
Extend the segment CD out until it intersects the third circle at point-E, forming radius
DE whose length is the square root of three times r.
It's at this point that Rene gets distracted, which isn't a surprise, as Mensa-ites often
notice things that are virtually invisible to ordinary mortals.
Rene focuses his attention on line segment CE, whose length is (the square root of two
plus the square root of three) times r...
...and the perpendicular line segment AC, whose length is r.
These line segments form two adjacent sides of a rectangle...
...whose area happens to be (the square root of two plus the square root of three) times
Rene calls this rectangle, Gaddy's Rectangle.
He then asserts that this "intermediate step" in Gaddy's construction should have the same
area as the beginning circle.
Unfortunately, Rene doesn't explain WHY he thinks this rectangle and the "circle in
question" should have equal areas.
He just arrives at the brilliant deduction that the longer side of the rectangle should
have a length of pi times r.
However, no part of this construction that we've seen so far would indicate that the
length of line segment CE should be equal to pi or anything except (the square root
of two plus the square root of three) times r.
There are no definitions, theorems or postulates that we can use to justify Rene's assumption
that the longer side of this Gaddy Rectangle has a length of pi.
Rene simply jumps to an erroneous conclusion.
But, the fact that Rene didn't understand any of Gaddy's math did not stop him from
offering his own "arithmetic" proof that this so-called Gaddy's pi is correct.
He does this by using a computer program that generates the area of the Gaddy rectangle
and compares it to the area of the circle using traditional pi and the new and improved
If pi equals (the square root of two plus the square root of three), the circle then
has "about" the same area as the rectangle, identical to only seven significant digits
using Rene's computer.
But, using traditional pi to calculate the area of the circle, the absolute difference
in areas gets larger as the number he uses for the circle's radius gets larger.
Hence, in Rene's vivid imagination, this proves Gaddy's pi is correct.
Rene doesn't realize that you cannot justify an incorrect value by simply demonstrating
how you could misuse that value.
Rene's "arithmetic" proof is no more valid than if he had used a completely different
rectangle to derive his value of pi.
Of course, it's odd that a Mensa-ite wouldn't realize that the ratio of the areas should
remain constant regardless of what value gets assigned to the radius.
Especially, since Rene gets excited over the fact that the tangent between the diagonal
and the base of Gaddy's rectangle remains constant regardless of the value of the radius.
Apparently Rene has never heard of proportional rectangles.
Regardless of what scalar you multiply the length and width of a rectangle by, that scalar
drops out when you take the ratio of the sides.
And the diagonals of proportional rectangles always have congruent angles
to their respective sides.
So, is there anything earth shattering in the part of Gaddy's construction
that we haven't seen yet?
Stick around and find out.
In part-two of this riveting investigation, we will finish Gaddy's construction and explain
why pi could not possibly equal (the square root of two plus the square root of three).