The main purpose of this book is to provide an ample store of such questions as are best suited to drill in elementary arithmetical operations and to oral practice in analysis and reasoning. That such training is most effectively given by means of questions easy enough to be worked mentally is an accepted opinion among experienced teachers. Accordingly, mental arithmetic fills a large place in courses of arithmetical study, and a separate book is often found to be a convenience, if not a necessity. It is not, however, as a substitute for written arith metic, nor yet as a distinct subject set apart from the other, that mental arithmetic will best serve its purpose for the two subjects are really one, differing chieﬂy in method of expression, and yielding their best fruits when both are applied to the same topics at the same time. Hence men tal arithmetic is an appropriate and useful study at all stages of the pupil's progress; and a book on the subject is not properly one member in a graded series of text-books, but is rather to be regarded as a companion to them, affording a parallel enlargement of the whole course.