The problem of Apollonius in the Urbino School (Q6651257)

One can begin with author's description:\N\N``During the Renaissance, several scholars worked to revive the contents and methods developed by the ancient Greek mathematicians. They began their research by studying the Latin editions of the Greek classics. The problem of Apollonius is a significant case study that sheds light on the recovery and re-appropriation of the solution methods employed by Greek mathematics. In this article, I will explore both the manuscript sources and the printed editions used by the Urbino School (Federico Commandino and Guidobaldo del Monte) to solve the problem of Apollonius.''\N\NApollonius of Perga (c. 240 BC--c. 170 BC) have formulated and solved the following problem: ``Given in position any three points, straight lines, or circles, to draw a circle through each of the given points -- if there be given any -- and tangent to each of the given (straight or circular) lines''. That is, ``finding one or more circles that are tangent to three given objects in a plane, which could be a point, a line, or a circle of any size'', but now the case of circles in this problem, i.e., the problem on ``constructing circles tangent to three given circles in a plane'', is called the problem of Apollonius.\N\NSince the original work of Apollonius is lost, the author gives the attention to Pappus' \textit{Collection}, where the problem of Apollonius was described. Federico Commandino (1506--1575) translated this Collection into Latin, ``faced the three-circle problem'', and involved Ettore Ausonio for solving the Apollonian problem, because this ``not adequately carried out by Pappus''. One can note the following sections:\N\N-- The Latin edition of Pappus' \textit{Collection}.\N\N-- The Latin edition of Proposition IV.7.\N\N-- Proposition IV.8 and the commentaries by Commandino.\N\N-- Propositions IV.9-10: the problem of the three circles.\N\N-- Guidobaldo del Monte and the Apollonian problem.\N\N-- Guidobaldo, Galileo, and the problem of Apollonius.\N\NIn this paper, the main discussion is given with auxiliary explanations, citations, and figures. Finally, the author notes that ``the problem of Apollonius represents both aspects of the rediscovery of ancient geometry and is an emblematic case study that demonstrates the importance of mathematical humanism in the development of modern science''.