Foreword by Maro K. Papathanasiou , Emeritus Professor of the Department of Mathematics of the National and Kapodistrian University of Athens.

This volume contains part of the second, third and fourth books of the "Synagogue" (the first book and part of the second have been lost). The second book deals with the methods used by Apollonius for performing multiplications of many and large numbers. The third book deals with finding two means of analogues for given two lines, the theory of means, exposes some "paradoxes", and finally gives a completely different from Euclid 's method of inscribing the five regular polyhedra on a sphere. In the fourth book, a generalization of the Pythagorean theorem is formulated, he deals with a passage called "arbelos", he presents earlier and his own solutions of the problem of squaring the circle, he describes the properties of a spiral circumscribed by a sphere, and finally he presents earlier and his own solutions of acute angle trisection problem.