Bing's Circular Square

Bing's Circular Square

Bing's Circular Square, named after Edward Bing, is a set square with the legs at the right angle being in a 44:23 ratio. It can be used for a simple approximative squaring of the circle.

Squaring of the circle

Squaring of the circle

Pick an arbitrary point $P$ on the circle. Then place the set square in such a way that the vertex at its smallest angle lies on $P$ and its longest side (hypotenuse) goes through the circle's center, that is it lies on a diameter of the circle. Now the other leg on the angle in $P$ intersects the circle in a second point $Q$ and the segment $PQ$ is the side of a square with approximately the same area as the circle. The deviation from the exact area is ${\tfrac {1}{1761}}\approx 0.000567859$.

Derivation of the ratio

The general idea is to determine the size of the angle $\alpha$ in $P$, such that $a=PQ$ will be the side of a square with the same area as the circle (see drawing). Then use this angle to construct a set square.

The desired equality in areas yields $a^{2}=\pi r^{2}$ and due to Thales's theorem the triangle $\triangle PQR$ has a right angle. Therefore $a=2r\cos(\alpha )$ and plugging that into the earlier equation yields $4r^{2}\cos(\alpha )^{2}=\pi r^{2}$. Hence $\cos(\alpha )={\tfrac {\sqrt {\pi }}{2}}$ which yields $\alpha =\arccos({\tfrac {\sqrt {\pi }}{2}})\approx 27.5971^{\circ }$

However to construct the set square one doesn't use the angle $\alpha$ directly but rather the ratio of the two legs at the right angle. That ratio is $\tan(\alpha )$. Using the Pythagorean trigonometric identity together with $\cos(\alpha )^{2}={\tfrac {\pi }{4}}$ from before yields $\sin(\alpha )^{2}={\tfrac {4-\pi }{4}}$ and hence $\tan(\alpha )^{2}={\tfrac {4-\pi }{\pi }}$. So $\tan(\alpha )={\sqrt {\tfrac {4-\pi }{\pi }}}\approx 0{,}522723200877063$ and approximating this value by a continued fraction results in $[0;1,1,10,1,1,126,3,45]$. The sequence of fractions associated with the development of this continued fracton is ${\tfrac {0}{1}},{\tfrac {1}{1}},{\tfrac {1}{2}},{\tfrac {11}{21}},{\tfrac {12}{23}},\mathbf {\tfrac {23}{44}} ,{\tfrac {2910}{5567}},{\tfrac {8753}{16745}}$ and ${\tfrac {396795}{759092}}$ with the fraction ${\tfrac {23}{44}}=0.52{\overline {27}}$ already yielding a good approximation for $\tan(\alpha )$.

History

The set square was devised by the Russian Edward Bing, who was the technical director of the Waggon Works in Riga.Three examples of the set square were part of an exhibition at the Science Museum in London in 1876, one made out of metal and two out of wood.

In 1877 Bing published an article on his set square in the German engineering journal Zeitschrift des Vereines deutscher Ingenieure under the title Der Kreiswinkel. The German name for his set square was derived from this and at the end of the 19th century it was manufactured out of ebonite in four different sizes by the German company Karl Schleicher und Schüll in Düren.

The idea of determining the angle $\alpha$ and potentially constructing a set square based on it occurred several times independently in literature. Aside from Bing's article in a German journal there were also articles in Nature in 1914 and 1918 and in the Mathematical Gazette in 1961.

External links

• Bing's Circular Square – interactive Illustration