Late of the Royal Laboratory Department, Woolwich, Professor of Mathematical Drawing to the principal establishments DRDICATED BY EXPRESS PERMISSION TO H. R. H. PRINCE ALBERT. SECOND EDITION, REVISED AND CORRECTED. LONDON: SIMPKIN AND MARSHALL, STATIONERS' COURT; RELFE & FLETCHER, CLOAK LANE; ROWNEY, DILLON, & ROWNEY, к THE first edition of the present work, consisting of fifteen hundred copies, having been disposed of before this went to press, and a second edition being loudly called for in many respectable quarters, it is presumed that the efforts of the Author to satisfy the public have been successful. The work, originally, was designed as a class book for a very numerous, intelligent, and influential body of persons-the members of Mechanics' Institutions; and, to their honour be it spoken, that upon the manuscript being submitted to the notice of Prince Albert, His Royal Highness was graciously pleased to signify his permission that the work should be dedicated to himself. The Author has had the pleasure and advantage of knowing that his little book has been introduced into the principal Mathematical Establishments of this country, including those specially designed for the in struction of the Nobility and Gentry, intended for Officers of the Army and Navy: it has also been ordered by the Lords of the Admiralty, and by the Honorable Board of Ordnance, for the education of their apprentices, and it has been honored with the most favorable critiques of the public press. In prefacing the second edition, it may be stated that scientific knowledge is constantly progressing among all classes in this country: to provide suitable elementary books, therefore, to further this object, is a task of weighty responsibility; and, upon no branch of science is a pioneer of this character more needed than the extensive subject of PRACTICAL GEOMETRY. The author of the following pages does not presume to rank himself with writers on Mathematical science, still, he hopes that the long experience of thirty years in almost every branch of mathematical drawing, adapted to Military and Civil Engineering, has enabled him to treat his subject so as to remove difficulties which have hitherto discouraged the student, and rendered the exertions of the teacher needlessly laborious; and whilst his work is intended, as originally, to meet the growing demands of scientific institutions, he trusts it will not be found unworthy the perusal of the private gentleman, nor unadapted to the studies and tastes of ladies. Woolwich, April, 1846. PRACTICAL GEOMETRY. INTRODUCTION. THE word Geometry is composed of two Greek roots, signifying "the earth," and " measure," and if the statements of Herodotus be true, that geometry took its birth from the annual overflowing of the Nile, and the consequent re-measurement of land; the term applied to this process was literally correct. That Geometry in some shape existed in the earliest ages of the world, can easily be conceived, as it would have been impossible for the ancients to have constructed edifices, except upon geometrical principles; besides which, on some of the most antique relics, astronomical emblems have been traced, which prove that geometry at an early period was applied, though crudely, to astronomy; and as it is quite clear that most departments of the exact sciences are up to this day entirely dependent upon geometry for their truth and usefulness, it is likely that it, from the first, advanced as rapidly as the limited facilities of the ancients would admit; till at the time of Euclid, who lived about 277 years before Christ, it had arrived to so great a state of perfection, that upon his "data" are based all the great scientific truths of the present day. Geometry is the science of magnitude, in its threefold properties; extension of length-extension of surface and extension of solidity; and the forms of objects, superficial and solid, can be so represented by its aid, as to convey the most correct ideas of their localities, bulk, and proportions; and upon it depend, |