From the Preface. The objects of this Manual are- First, to furnish to students full explanations of all the difficulties usually met with in the practice of computation by logarithms; Secondly, to set before them compact and orderly forms of arrangement of the computations; and, Thirdly, to provide instructors with an abundant collection of examples, progressively arranged, and covering all the points in regard to which mistakes are commonly made. The Manual is the result of many years' experience of the difficulties met with in endeavoring to teach young students orderly and correct, as well as intelligent methods; and whatever may be thought of the success with which these difficulties have been treated, it is believed that not many of them have been overlooked. The rather elaborate subdivision of some of the topics in the third chapter results from actual experience of the perplexity occasioned in the minds of even pretty good students by the varying signs which characteristic and mantissa and their multipliers assume in different problems. It is strongly recommended that the teacher insist upon the pupil's following, in all written exercises, the form given in the last solved example under each head. Orderly arrangement is almost indispensable to correct and rapid work, and quite indispensable to that thorough revision by the teacher, without which written exercises are, for most pupils, of little value.