*A good pi day to all readers! I hope that the following post, that I cannot review after the first writing, could be interesting to all!*The technique used by the ancient Greek for their geometric constructions was called "ruler and compass". In this way you can build a lot of regular polygons, for example, but there are three problems that are impossible unless you use different techniques: the angle trisection, doubling the cube, squaring the circle.

In particular for the squaring, it is easy to calculate the relation between the radius $r$ of the circke and the side $l$ of the square with the same area of the starting circle: \[L = \sqrt {\pi} r \] Now, since $\pi$ is a transcendental number, the formula above is the simplest representation of the impossibility of squaring the circle using ruler and compass: with these devices you can treat rational and irrational numbers, such as $\sqrt{2}$ (in this case simply draw a square of side 1).

So, using these two tools it is possible to obtain an approximate construction and, therefore, a corresponding approximate value for $\pi$: during the XX century there are produced a lot of approximations, for example by

**CD Olds**(1963),

**Martin Gardner**(1966),

**Benjamin Bold**(1982). They are all variations of the geometric construction discovered by

**Srinivasa Ramanujan**in 1913, that approached $\pi$ with the fraction \[\frac{355}{113} = 3.1415929203539823008 \dots\] right up to the sixth decimal place.

In 1914, Ramanujan discovered a more accurate approximation (to eight decimal places), always using ruler and compass: \[\left (9^2 + \frac{19}{22}^2 \right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252 \dots\] If however, to compass and ruler we add a bit of mechanics, we can get an interesting way to square the circle (with a subsequent measure of $\pi$!). Take a circle of radius $r$ and let's roll with constant speed along its tangent line. The point of tangency $N$ describe a curve called a cycloid.

Now you can construct a second cycloid using the projection of the point $M$ on the tangent line. The intersection $B$ between the diameter $NN'$ with this second cycloid will provide a segment $NB$ whose length is equal to $\sqrt{\pi} r$, which is the side of the square with the same area of the starting circle. In this way, using a bit of algebra, is it also possible to determine the value of $\pi$.

But this construction is not simply geometric, but we can also do it really, perhaps with a small brass disc as suggested by

**August Zielinski**, the proponent of the method of squaring just now briefly described.

Obviously in this case the precision of the value of $\pi$ depends on the accuracy of the devices used.

**Papers by Ramanujan**:

Squaring the circle,

*Journal of the Indian Mathematical Society*, V, 1913, 132

Modular equations and approximations to $\pi$,

*Quarterly Journal of Mathematics*, XLV, 1914, 350 – 372

Zielinski A. (1875). Quadrature of the Circle, The Analyst, 2 (3) 77. DOI: http://dx.doi.org/10.2307/2635871 (archive.org)

**Reading also**:

Crop Circles and More | McTutor |

*Squaring the circle: a history of the problem*by

**E. W. Hobson**

*Wallis product*, discovered more or less by an accident by the mathematician

**John Wallis**, as he was trying to calculate the area of a circle. \[\prod_{n=1}^\infty \left ( \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} \right ) = \frac{2}{1} \cdot \frac{2}{3} \cdot {4}{3} \cdot \frac{4}{5} \cdots = \frac{\pi}{2}\] Wallis presented his formula in his most famous book,

*Arithmetica infinitorum*(arithmetic of the infinite) of 1665, essential for the

**Isaac Newtons**'s training, who became his successor as a landmark of British mathematicians.

However, the history of the formula began in Italy in 1632 with the publication of

*Lo Specchio Ustorio, overo, Trattato delle settioni coniche*(the burning glass, or a treatise about conic sections) where the mathematician

**Bonaventura Cavalieri**calculates the area under "parables" of type $y = x^n$. Cavalieri proceeded by calculating the area between the axes and curves of type \[y = \left ( 1-x^2 \right )^0, \; y= \left ( 1-x^2 \right )^1 \; y= \left ( 1-x^2 \right )^2\; y= \left ( 1-x^2 \right )^3, \; \cdots\] obtaining \[x,\] \[x - \frac{1}{3} x^3,\] \[x - \frac{2}{3} x^3 + \frac{1}{5} x^5,\] \[x - \frac{3}{3} x^3 + \frac{3}{5} x^5 - \frac{1}{7} x^7,\] \[x - \frac{4}{3} x^3 + \frac{6}{5} x^5 - \frac{4}{7} x^7 + \frac{1}{9} x^9,\] \[\cdots\] Since the equation of the circle of radius 1 is $y = \left (1-x^2 \right)^{1/2}$, the problem reduces to determining the expression corresponding to the exponent $\frac{1}{2}$ between $x$ and $x - \frac{1}{3} x^3$.

Unable to determine the expression, Wallis completed a series of numerical calculations that led him eventually to the formula that bears his name. A more or less simple proof of the formula passes through the definition of three new series and the numerical calculation of the area of rectangles in the Cartesian plane: already just looking at the picture below is easy to understand how these series, approximating the rectangles, to the limit they approach the arc of a circle, so converging to the value of $\frac{\pi}{2}$.

A more rigorous demonstration, however, that takes the road of the proof of the formula of Cavalieri, passes through the integral of the breast \[\int_0^\frac{\pi}{2} \sin^n x dx\] It is also possible, using suitably the Dirichlet eta function, to determine a link between the Riemann zeta and the formula of Wallis, which is also quite obvious because the Riemann zeta is linked to pi.

Wildberger N.J. (2002). A New Proof of Cavalieri's Quadrature Formula, The American Mathematical Monthly, 109 (9) 843. DOI: http://dx.doi.org/10.2307/3072373

Young R.M. (1998). Probability, pi, and the Primes: Serendipity and Experimentation in Elementary Calculus, The Mathematical Gazette, 82 (495) 443. DOI: http://dx.doi.org/10.2307/3619891

Wästlund J. (2007). An Elementary Proof of the Wallis Product Formula for pi, The American Mathematical Monthly, 114 (10) 914-917. DOI: http://dx.doi.org/10.2307/27642364 (pdf)

Young R.M. (1998). Probability, pi, and the Primes: Serendipity and Experimentation in Elementary Calculus, The Mathematical Gazette, 82 (495) 443. DOI: http://dx.doi.org/10.2307/3619891

Wästlund J. (2007). An Elementary Proof of the Wallis Product Formula for pi, The American Mathematical Monthly, 114 (10) 914-917. DOI: http://dx.doi.org/10.2307/27642364 (pdf)

**Reading also**: Mathworld | The world of $\pi$ | Math fun facts
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