AN INTRODUCTION TO MENSURATION AND PRACTICAL GEOMETRY. BY JOHN BONNYCASTLE, OF THE ROYAL MILITARY ACADEMY, WOOLWICH. TO WHICH ARE ADDED, A TREATISE ON GUAGING: AND ALSO THE MOST IMPORTANT PROBLEMS IN MECHANICS. BY JAMES RYAN, Author of a Treatise on Algebra, the New American Grammar of Astronomy, Philadelphia: KIMBER & SHARPLESS, No. 8 SOUTH FOURTH ST. STEREOTYPED BY J. HOWE. ....... 1835. ENTERED according to the Act of Congress, in the year 1833, by KIMBER & SHARPLESS, in the Clerk's Office of the District Court of the Eastern District of Pennsylvania. TO THE LONDON EDITION. THE ART OF MEASURING, like all other useful inventions, appears to have been the offspring of want and necessity; and to have had its origin in those remote ages of antiquity, which are far beyond the reach of credible and authentic history. Egypt, the fruitful mother of almost all the liberal sciences, is imagined likewise to have given birth to GEOMETRY Or MENSURATION; it being to the inundations of the Nile that we are said to be indebted for this most perfect and delightful branch of human knowledge. After the overflowings of the river had deluged the country, and all artificial boundaries and land-marks were destroyed, there could have been no other method of ascertaining individual property, than by a previous knowledge of its figure and dimensions. From this circumstance, it appears highly probable, that Geometry was first known and cultivated by the ancient Egyptians; as being the only science which could administer to their wants, and furnish them with the assistance they required. The name itself signifies properly the art of measuring the earth; which serves still further to confirm this opinion, especially as it is well known that many of the ancient mathematicians applied their geometrical knowledge entirely to that purpose, and that even the Elements of Euclid, as they now stand, are only the theory from whence we obtain the rules and precepts of our present more mechanical practice. But to trace the sciences to their first rude beginnings, is a matter of learned curiosity, which could afford but little gratification to readers in general. It is of much more consequence to the rising generation to be informed that, in their present improved state, they are 156370 exceedingly useful and important. And in this respect, the art I have undertaken to elucidate is inferior to none, arithmetic only excepted. Its use in most of the different branches of the Mathematics is so general and extensive, that it may justly be considered as the mother and mistress of all the rest, and the source from whence were derived the various properties and prin ciples to which they owe their existence. As a testimony of this superior excellence, I need only mention a few of those who have studied and improved it; in which illustrious catalogue we have the names of Euclid, Archimedes, Thales, Anaxagoras, Pythagoras, Plato, Apollonius, Philo, and Ptolemy, amongst the ancients: and Huygens, Wallis, Gregory, Halley, the Bernouillies, Euler, Liebnitz and Newton, amongst the moderns; all of whom applied themselves to particular parts of it, and greatly enlarged and improved the subject. To the latter especially we are indebted for many valuable discoveries in the higher branches of the art; which have not only enhanced its dignity and importance, but rendered the practical application of it more general and extensive. The degree of estimation in which the art was held by these, and other eminent characters, will, in general, it is apprehended, be thought a sufficient encomium on its merits. But, for the sake of young people, and those of a confined education, it may not be amiss to give a few more instances of its advantage, and show that its importance in trade and business is not inferior to its dignity as a science. Artificers of almost all denominations are indebted to this invention for the establishment of their several occupations, and the perfection and value of their workmanship. Without its assistance, all the great and noble works of Art would have been imperfect and useless. By this means the architect lays down his plan, and erects his edifice; bridges are built over large rivers; ships are con structed; and property of all kinds is accurately measured, and justly estimated. In short, most of the elegances and conveniences of life owe their existence to this art, and will be multiplied in proportion as it is well understood, and properly practised. From this view of the subject, it is hardly to be accounted for, that, in a commercial nation, like our own, an art of such general application should have been so greatly neglected. Mechanics of all kinds, it is well known, are but ill acquainted with its principles; and those who have been the best qualified to afford them any assistance, have thought it beneath their attention. Till within a few years past, there could not be found a regular treatise upon this subject in the English language. Some particular branches, it is true, had been greatly cultivated and improved; but these were only to be found in their miscellaneous state, interspersed through a number of large volumes, in the possession of but a few, and in a form and language totally unintelligible to those for whom they were more immediately necessary. Dr. Hutton was the first person, in this country, who undertook to collect these scattered fragments, and to treat of the subject in a scientific, methodical manner. A small treatise by Hawney, and some others of little note, had indeed been long in the hands of the public; but these were extremely defective, both in matter and method; neither the principles nor practice of the art being properly or clearly explained. Before the publication of the treatise above mentioned, Mr. Robertson's may be considered as the only book, of any value, that could be consulted, either by the artisan or mathematician; and had he given the theory as well as the practice of the art, and divested his rules and examples of their algebraical form, there would have been no want of any other elementary treatise. To these two writers I am greatly indebted for many |