IN this Edition considerable alterations have been made in chapter xii. In particular, the proof of the theorem that every integral equation has a root has been amplified, and also illustrated by graphical considerations. An Appendix has been added dealing with the general algebraic solution of Cubic and Biquadratic Equations; with the reducibility of equations generally; and with the possibility of solution by means of square roots. As the theorems established have interesting applications in Ele mentary Geometry, it is believed that they may find an appropriate place in an Elementary work on Algebra.