“NUMBER,” infers “Recorde,” in his “Whetstone of Witte,” “is the onelie thing (almost) that seperateth man from beastes. Hee therefore that shall contempne numbre, he declareth himselfe as brutishe as a beaste, and. unworthy to be counted in the fellowshippe of men. But I truste there is no man so foule ouerseene, though manie right smallye do it regarded”—(De Morgan, “English Mathematical and Astronomical Writers.”) We have done with the miserable mercantile compendiums founded on Cocker, which De Morgan condemned, and have had, since his own “Arithmetic” appeared, many works of high value. Mr. Muir's work is worthy of taking place with these. His aims are high—mathematical accuracy, rational treatment, the presentment of essentials, with the accessories in due subordination, the production of a work suited both for mental training and as a preparation for the practical business of life. There is perhaps matter given more suitable for the use of teachers than of pupils—that is, for a school book we think much might be more concisely put. Persons taking up the subject at a more advanced age may find this fulness of explanation very valuable. The exercises are good and varied, and there is a chapter containing notices of books for future reading.