In publishing the following work, my principal intention is to explain difficulties, which may be encountered by the student on first reading the Principia, and to illustrate the advantages of a careful study of the methods employed by Newton, by showing the extent to which they may be applied in the solution of problems. I have also endeavoured to give assistance to the student who is engaged in the study of the higher branches of Mathematics, by representing in a geometrical form several of the processes employed in the Differential and Integral Calculus, and in the analytical investigations of Dynamics.<br><br>In my version of the first section and the beginning of the second I have adhered as closely as I could to the original form; and, in the cases in which sections have been inters polated, or the form of demonstration changed, I have indicated such changes and interpolations by brackets.<br><br>Although it is generally advisable not to deviate from Newton's words in the demonstrations of the Lemmas, yet in many cases, I suppose, purposely, he expressed himself very concisely, as in Lemmas IV. and X.; and he was contented with simply giving the enunciation of Lemma V.; in these cases, therefore, interpolations are made which, I believe, are in accordance with Newton's plan of demonstration.<br><br>Throughout the Problems and Theorems which depend upon the sixth proposition, the variations are replaced by equations; by this method of treating the subject, I conceive that clearer ideas of the meaning of each step are obtained by the student.