Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/39

${\frac {1440}{458{\frac {4}{9}}}}=3{\cdot }1410\ldots$, whereas $\textstyle 3{\frac {1}{7}}=3{\cdot }1428$, $\textstyle 3{\frac {10}{71}}=3{\cdot }1408\ldots$ were the values given by Archimedes. From these limits he chose ${\frac {1440}{458{\frac {1}{3}}}}$ or $\pi =3{\cdot }1418\ldots$ as the mean result.

During the period of the Renaissance no further progress in the problem was made beyond that due to Leonardo Pisano; some later writers still thought that 3 1/7 was the exact value of $\pi$. George Purbach (1423—1461), who constructed a new and more exact table of sines of angles at intervals of 10′, was acquainted with the Archimedean and Indian values, which he fully recognized to be approximations only. He expressed doubts as to whether an exact value exists. Cardinal Nicholas of Cusa (1401—1464) obtained $\pi =3{\cdot }1423$ which he thought to be the exact value. His approximations and methods were criticized by Regiomontanus (Johannes Müller, 1436—1476), a great mathematician who was the first to shew how to calculate the sides of a spherical triangle from the angles, and who calculated extensive tables of sines and tangents, employing for the first time the decimal instead of the sexagesimal notation.

The fifteenth and sixteenth centuries

In the fifteenth and sixteenth centuries great improvements in trigonometry were introduced by Copernicus (1473—1543), Rheticus (1514—1576), Pitiscus (1561—1613), and Johannes Kepler (1571—1630).

These improvements are of importance in relation to our problem, as forming a necessary part of the preparation for the analytical developments of the second period.

In this period Leonardo da Vinci (1452—1519) and Albrecht Durer (1471—1528) should be mentioned, on account of their celebrity, as occupying themselves with our subject, without however adding anything to the knowledge of it.

Orontius Finaeus (1494—1555) in a work De rebus mathematicis hactenus desiratis, published after his death, gave two theorems which were later established by Huyghens, and employed them to obtain the limits 22/7, 245/78 for $\pi$; he appears to have asserted that 245/78 is the exact value. His theorems when generalized are expressed in our notation by the fact that $\theta$ is approximately equal to $\textstyle (\sin ^{2}{\theta }\tan {\theta })^{\frac {1}{3}}$.