Squaring the Circle and Doubling the Cube
Of ferman: Fernando Mancebo Rodriguez--- Personal page. ----Spanish pages

You can see many of my works, in the following pages:

Email: ferman25@hotmail.com
Email: ferman30@yahoo.es

Squaring the Circle and Doubling the Cube

Fernando Mancebo. April 1, 2023 Fig.1

Friends, I am going to expose this resolution of the squaring of the circle, for which it will be necessary at the same time to solve the other problem of doubling the cube, since for this squaring it is necessary to apply the segment cube root of 2.
However, we will first proceed to squaring the circle by applying the cube root of 2, to study, adjust numerically and better understand the squaring procedure.
Then, at the end, the extraction of the cube root segment of 2 is put, to build the squaring of the circle in practice, with a ruler and compass. As seen in figure 1, the squaring will be carried out by means of the construction of two squares inscribed in the circle, which must be properly aligned to achieve the 8 anchor points (yellow), which together will give us the side (Sqrt. Pi) and wanted square (blue).

The theory and mechanics of drawing is very easy and simple to carry out.

As we will see, used square root of 2 to build the squares inscribed in the given circle; and we use the segment cube root of 2 to align these two constructed inscribed squares, and mark the anchor points for the final square of the quadrature. Fig.2

In this first drawing of figure 2, we expose the initial circle to be squared with its various diameters (4), and then we mark the first inscribed square. Fig.3

In this drawing Fig.3, we transfer the cube root segment of 2 (A-c,), previously obtained, over the center c, and then with the compass we make an arc until cutting the square inscribed in c'.
And we proceed to draw the line A-c'-p, which will mark the first anchor point p1.
Note.- For a better adjustment (especially mathematical) we can take the arc c'-p8, to mark the initial point p1. of the second inscribed square. Fig.4

Now, Fig. 4, we proceed with the compass and square root of 2, to mark and draw the second square starting from the previous point found p1, and marking the other three points of the square p3, p5, p7.
And finally with the rule, we unite and extend the anchor points properly as shown in figure 4/2 to build the desired square of the squaring of the circle.
* In the following drawing some parameters and values of segments and angles are provided. Doubling the cube to get cube root of 2

* To double the cube, it is very convenient to use a compass with two very fine needles to be able to fix it well to the paper and so that it does not slip in the measurements and turns. Fig.1

To double the cube, we have the following parameters:
1.- Firstly, the radius = 1 of the circumference and its diameter 2, whose cube root that we seek is 1.259921...
2.- Secondly, and starting from the principle (A) of the diameter of the circumference, we draw a chord of circumference C (Section AB, which in drawing practice, and to facilitate the work, will be replaced by the ruler fig. 2). Fig.2

3.- Immediately we see that this chord C has its projection on the diameter in the segment (p), which forms the third element necessary to find the cube root of 2 (C).
Well then, if we observe and work with two of these three elements, the chord (C) and its projection segment (p), we build a formula that is basic and essential for our research: C x p = (from 0 to 4)
But pay attention! When this formula ( C x p) gives us the radius of the circle 1 (C x p = 1, third element), then we measure C and see that it is exactly the cube root of 2, that is, the segment we were looking for. (1.259921... x 0.79370052.. = 1).
Therefore, we already observed that there is an interrelation or quadrature by means of segments and formulas between the radius (or diameter) of the circumference, with a cube root of 2, and its projection on the diameter.
At this point we can already begin to feel some restlessness and intrigue:
Will we be able to obtain or "hunt" the elusive segment C, that is, cube root of 2?
Well, let's not worry, the answer is yes, even fun to do.
But, how to do it?
Well, using the component elements of the previous formula, C.p = 1, but using ruler and compass instead of numbers and quantities.
- The chord segment will be replaced by the ruler with which it rests at the beginning of the diameter (A) until cutting the circumference at point (B).
- The projection segment on the diameter (p) will be used as an arc (arc p) that will intersect (arc.r)
- The radius (1) will be used to create another arc of connection and measure (arc.r) that will be the unit of measure and future side of the cube to be duplicated. Next Fig. 3 Fig.3

In this section we should already forget a bit about the measurements and numerical verifications: Let's take the compass and the ruler, and proceed to draw on a circle of radius 1, with its centre and diameters already marked.
- The first thing to build is an arc of radius = 1, (with centre at point A) and from the centre of the circumference to the left, which serves as a guide, measure and support for the other two basic elements that we are using.
- Now, we take the rule and at the beginning of the diameter (point A), we cut with it the circumference at the test point (B), creating a space or segment from this point to the vertical diameter.
- To this segment or distance (a-a'), we measure it with the compass, and we translate this measurement on the horizontal diameter creating the segment (a-a'), at which point (O) we begin the verification of the confluence of elements with the compass.
Logically, the first tests will fail and there will be no coincidence between the rule and the cut between the arcs, (cut arc r - arc p).
So we move the ruler in the right direction and try again the squaring or confluence of elements.
By moving the ruler in the correct direction and carrying out the above checks, after 3-5 attempts, the quadrature will have been achieved and we will be at the point (q) of confluence of these three elements.
At that time, we mark point B on the circle and we can definitely draw the segment AB = C which is the cube root of 2.

Explanation of the process:

As we have seen, to square the circle we need the cube root of 2, and therefore we must first find this segment (cubic root of 2)
However, this double process results in the discovery of two unknown properties of the circumference, and therefore a good contribution to the knowledge of this geometric figure. Doubling the cube:

The first (by doubling the cube) we discover that there is a hidden parameter in the circumference, which is the cube root of 2.
And I call it hidden because to discover it is necessary to do it through mathematical operations.
This parameter or segment (C) inscribed in the circumference consists of a chord of the same, which if we multiply it by its projection on the diameter (placing on one of the cuts of this circumference chord) gives us its radius 1.
And then we will see that this chord or segment measures exactly the cube root of 2 (1.259921.....
Therefore finding this chord we find the cube root of 2.
And as we have seen in the previous explanation, it is very easy to achieve by means of a single expert measurement with the compass.
And I say expert because the better we know how to handle the compass and the greater its reliability, the faster and more accurate the measurement will be.
However, and as we have seen that the method is correct and exact, the process will also be correct and exact. Squaring the circle

In squaring the circle we also discover another (until now) unexpected property or characteristic of the circumference:
And it is that the cube root of 2 is mathematically and geometrically interconnected with the circumference.
In such a way that if we place it properly on the circumference or circle to be squared, it quickly produces a situation and alignment of two squares inscribed in this circumference, and whose anchor points to it define a superior square whose side is the square root of Pi, that is, the square sought for the squaring of the circle. Curiosity: (drawing

I often proceed to squaring the circle, doubling the cube beforehand to find the cube root of 2.
For this, I use a compass with two fine needles, and a sheet of paper as a ruler, since this prevents the thickness of a ruler from preventing the perfect location of the compass needles at the appropriate points.
And it is precisely this work of doubling the cube that I like the most about the process, since in less than a minute the only measurement that is necessary to find the segment cube root of 2 is achieved.