What i can't understand is that I'm reading book " a History of Mathematics by boyer" and it says Archimedes made possible to construct a triangle equal in area to that of a circle by help of spirals. Etc.. And then it says " with a simple geometric transformation you can produce a square out of it. But, wait.
isnt the squaring of circle concidered impossible all the time. I am very confused. Plz help me understand.
On
We certainly can square the circle with the right devices.
Archimedes is better known for approximating the circumference rather than rendering an exact construction:
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes.[1] This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant".[2] Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that
223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).[3]
Cited references:
1. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka, p. 169.
2. Ibid., p. 170.
3. Ibid., pp. 175, 205.
The problem of "squaring the circle" without qualifications usually refers to doing it with only compass and straightedge, which is indeed impossible. On the contrary, solutions that use further tools not emulatable with compass and straightedge, like spirals, were known even in antiquity.