The Works of Euclid, Vol. 3

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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 bodies attributed to Hypsicles.

If I have appended the two books attributed to Hypsicles to the works of Euclid, it was to conform to established usage. I do not mean by this that the first book is not a precious monument of ancient geometry. As for the second, it is quite otherwise: the demonstrations of this book are incomplete, lacking in rigor and elegance; which leads me to believe that not only are these two books not by the same author, but also that one is much older than the other.

This volume contains a very large number of variants, more or less precious. I leave to the reader the task of evaluating them at their leisure.

Variant 4 of Proposition I of the eleventh book demonstrates in a simple and elegant manner that two straight lines cannot have a common part without coinciding. This proposition, which could be a corollary of Proposition XIV of the first book, is placed by Proclus among the axioms, with a demonstration similar to that of this variant, which I have not adopted.

Proposition XVII of the twelfth book, which is one of the most important in Euclid, had been regarded as incomplete until now, from the paragraph on page 196 to the corollary on page 205. I have shown in a note placed at the bottom of page 200 that this demonstration was complete in all its parts, and that all the confusion arose only from a poorly constructed figure.

One might perhaps say that Archimedes arrived more directly at the goal, which is to demonstrate the ratio of two spheres of unequal magnitude; this is very true. Indeed, Archimedes having demonstrated that spheres are equal to two-thirds of their circumscribed cylinders, it follows evidently from this that spheres are to one another as the cubes of their diameters.