Some apology is required for adding another to the long list of books on Trigonometry. My excuse is that during twenty years` experi- ence I have not found any published book exactly suiting the wants of my Students. In conducting a Junior Class by regular progressive steps from Euclid and Elementary Algebra to Trigonometry, I have had to ll up by oral instruction the gap between the Sixth Book of Euclid and the circular measurement of Angles; which is not satisfactorily bridged by the propositions of Euclid`s Tenth and Twelfth Books usually supposed to be learned; nor yet by demonstrations in the modern books on Trigonometry, which mostly follow Woodhouse; while the Appen- dices to Professor Robert Simson`s Euclid in the editions of Professors Playfair and Wallace of Edinburgh, and of Professor James Thomson of Glasgow, seemed to me defective for modern requirements, as not suciently connected with Analytical Trigonometry.