The present book is a translation into English of Elernenta CU'f'Varurn Linearurn-Liber Prirnus, written in Latin, by the Dutch statesman and mathematician Jan de Witt (1625-1672). Together with its sequel, Ele- rnenta CU'f'Varurn Linearurn-Liber Secundus, it constitutes the first text- book on Analytic Geometry, based on the ideas of Descartes, as laid down in his Geornetrie of 1637. The first edition of de Witt's work appeared in 1659 and this translation is its first translation into English. For more details the reader is referred to the Introduction. Apart from this translation and this introduction, the present work con- tains an extensive summary, annotations to the translation, and two ap- pendices on the role of the conics in Greek mathematics. The translation has been made from the second edition, printed by the Blaeu Company in Amsterdam in 1684. In 1997 the translator published a translation into Dutch of the same work, likewise supplied with an introduction, a summary, notes, and two appendices. This edition appeared as a publication of the Stichting Mathe- matisch Centrum Amsterdam. The present translation, however, is a direct translation of the Latin text.The rest of this work is an English version of the introduction, the summary, the notes, and the appendices, based on the Dutch original.