PREFACE.

Euclid lived in the time of Ptolemy Lagus, around the year 272 before the common era; Archimedes cited him in several of his books. When Ptolemy asked Euclid if there were not an easier way to learn Geometry than his own, Euclid replied that there was no royal road to arrive at this science. This is all that we know of Euclid; one does not even know what his homeland was.

Many geometers had appeared before Euclid. The first of the Greeks, Euclid gathered their works, put them in a suitable order, and provided unassailable demonstrations for what had not been demonstrated in a rigorous manner.

Euclid had composed a large number of works. The thirteen books of the Elements and the Data are the only ones that have reached us.

Euclid's Elements have always been regarded as the most perfect of all elementary books; they have been translated and commented upon in every language.

Cardano, in speaking of Euclid's Elements, expresses himself thus: Quorum inconcussa dogmatum firmitas, perfectioque adeo absoluta est, ut nullum opus jure huic aliud comparare audeas ; quibus fit ut adeo veritatis lux in eo refulgeat, ut soli hi in arduis quæstionibus videantur posse a vero falsum discernere, qui Euclidem habeant familiarem.

Pemberton informs us that he had heard Newton complain several times of having devoted himself entirely to the works of Descartes, and of other algebraists, before having studied and meditated upon the Elements of Euclid.

M. Lagrange, whose loss Europe laments and will long lament, often repeated to me that Geometry was a dead language; that he who