Squaring the Circle Using Modified Tartaglia Method

The paper presents a modified Tartaglia method. Tartaglia proposed a simple approach to perform an approximate quadrature of the circle. His construction results with the number pi=3.125. Using a similar construction as Tartaglia but with different proportions improves the accuracy of his method.

Introduction

Italian mathematician Niccolo Fontana Tartaglia (1500–1557) in his book [1], presented the following approximate squaring the circle. He started from a given square with diagonal. He transformed the square into the circle dividing its diagonal into ten equal parts; the corresponding circle as its diameter (d) has eight parts of the diagonal of the square. It can be seen as Tartaglia used a set square with the sides in proportion 8:10 or 4:5. This idea (using a set square) will be further developed in this short note.

Consider the square with the side of length a. Its diagonal is z=a√2. Tartaglia method gives the following relation πr^2=π〖(d/2)〗^2=π〖(0.8z/2)〗^2=π〖((0.8a√2)/2)〗^2=a^2. This condition that the areas of the square and the circle constructed in such way are equal results in the number π=3 1/8. Thus the approximation is the same as was obtained in ancient Babylon.

Method

The following question holds: is it possible to improve this method? The answer to this question is yes, it can be done better. Now the idea is that rather to scale the diagonal z by 0.8, lets try to find the scale factor x, which gives better approximation. For this purpose consider more general approach based on the following relation πr^2=π(d/2)^2=π(xz/2)^2=π((x a√2)/2)^2=a^2. From this equation the factor x is easily determined as x=√2/√π≈0.797884560802865…. Using the continued fraction, we have the following 19 first terms for x=[0;1,3,1,18,9,4,1,4,3,2,1,3,1,1,2,1,5,1,…]. The infinite continued fraction representation for x is very useful here. Using rational approximations to the number x it is possible to construct a set square which can be used to perform the approximate quadrature of the circle.

Results

The corresponding convergents of the determined continued fraction for x are presented in Table 1. The table also contains the approximate values for the number pi. By constructing the set square with sides in the proportion 35567099:44576748 the tool will do approximate quadrature of the circle with π≈3.141592653589795 where true π≈3.1415926535897931.

Table 1. Numerator, denominator, fraction and the approximation to the number pi.

Here can be mentioned that the angle corresponding to x is 38.5858260136047…degrees, i.e. x≈tan(38.5858260136047)= 0.797884560802865. In practice, it is better to use the right triangle with sides of the suggested proportions. The triangle is symmetrical in its purpose; for a given circle (square) determines the square (circle). A very similar idea was proposed by a Russian engineer Edward Bing around the year 1877 [2-5].

1. Tartaglia N (1556) General trattato di nvmeri:  et misvre Link: https://goo.gl/VPIBgG
2. Szyszkowicz M (2016) Krótka historia ekierki Binga (in Polish: A short history of Bing's set square) East European Scientific J (EESJ) 12: 2.
3. Szyszkowicz M (2016) Ahmes' method to squaring the circle: European Journal of Mathematics and Computer Science (EJMCS) 3.
4. Szyszkowicz M (2016) Ancient Egyptian Quadrature Executed Using A Set Square: Journal of Multidisciplinary Engineering Science and Technology (JMEST)  3.  Link: https://goo.gl/zkw6oz  
5. Bing E (1877) Der Kreiswinkel. Vermischtes: VDI-Z: Zeitschrift für die Entwicklung, Konstruktion, Produktion 21: 273-279.