The design of this work is twofold : 1. As a supplement to the author's large and advanced treatise, which does not contain sufficient primary work to meet the wants of young pupils. 2. As an introductory treatise to the science of numbers. It is especially designed to supply the requirements of primary and intermediate classes, and, at the same time, it presents much practical work of rare import to the advanced student. Bills and invoices of various forms for many departments of business constitute a special feature of the work. It is believed to possess superior merit on the following points: 1. In the arrangement and character of the mental exercises and the logical methods of mental training. 2. In the extent, variety' practical and scientific character of the problems. 3. In the elucidation of subjects. 4. In the philosophic solution of problems, by which system all the reasoning organs of the mind are expanded and the learner capacitated, not only to produce the results of problems, but to observe fine distinctions, reason logically, and deduce correctly; thus qualifying for a high plane of usefulness in various vocations of life.<br><br>The philosophic system is believed to be the most valuable improvement yet made to impart a thorough knowledge of the principles of numbers and capacitate the learner to utilize the same in the practical affairs of business life, and as it is the only natural system, it is destined at no distant day, to be adopted by all reasoning minds and efficient instructors.