PREFACE.

This third and final volume contains books XI, XII, and XIII of the Elements and the Data of Euclid, as well as the two books on the five Solids which are ascribed to Hypsicles.

I have added the two books of Hypsicles to the works of Euclid so as not to depart from ancient custom. Yet I would not deny that the former is to be commended as being a certain monument of ancient geometry; as for the latter, I confess that I feel quite otherwise. For the demonstrations of this book are incomplete, and are lacking in rigor and elegance; therefore, I judge that not only should these books not be ascribed to the same author, but also that one is much older than the other.

This volume contains very many various readings of greater or lesser value, which it will be possible for each person to weigh with an attentive mind.

The various reading of Proposition I of the eleventh book shows simply and elegantly that if two straight lines have a common part, they coincide with one another. This proposition, which could be a corollary to Proposition XIV of the first book, was placed by Proclus among the axioms with a demonstration similar to the demonstration of this various reading, which I have not admitted.

Proposition XVII of the twelfth book, one of those which is of the greatest importance, was held until now to be incomplete from the start of page 196 to the corollary on page 205. In the note which is at the bottom of page 200, I have shown that this demonstration is complete in all its parts, but that the figure is entirely disordered.

If anyone says that Archimedes arrived more directly at the goal, which was the discovery of the ratio of two spheres unequal in magnitude, I indeed concede it. For from the fact that Archimedes demonstrated that spheres are equal to two-thirds parts of their circumscribed cylinders, it is manifest that spheres are to one another as the cubes of their diameters.