from which attempts have been made to restore the lost originals. Two books on De Sectione Rationis have been found in the Arabic. The book on Contacts, as restored by Vieta, contains the so-called "Apollonian Problem": Given three circles, to find a fourth which shall touch the three.
Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent, during which they paused here and there to look around for details which had been passed by in the hasty ascent.
Among the earliest successors of Apollonius was Nicomedes. Nothing definite is known of him, except that he invented the conchoid ("mussel-like"). He devised a little machine by which the curve could be easily described. With aid of the conchoid he duplicated the cube. The curve can also be used for trisecting angles in a way much resembling that in the eighth lemma of Archimedes. Proclus ascribes this mode of trisection to Nicomedes, but Pappus, on the other hand, claims it as his own. The conchoid was used by Newton in constructing curves of the third degree.
About the time of Nicomedes, flourished also Diocles, the inventor of the cissoid ("ivy-like"). This curve he used for finding two mean proportionals between two given straight lines.