PREFACE. SCIENTIFIC PURSUITS have, in all ages, been considered of the greatest utility to mankind in general; and those individuals who have laboured successfully to elucidate the principles of Mathematics, or to simplify their appli eation to the Practical affairs of life, have always been ranked among the most useful and honourable members of society. Honourable indeed they must appear to all those who have learned to set due estimation on intellectual attainments; as, next to vital religion, they certainly constitute the true honour and dignity of man. Riches and titles may be our legal possessions, by birthright; however, in a meritorious point of view, they can scarcely be denominated our own; but the treasures of a well-stored mind can only be acquired by diligent study, and assiduous application; and are therefore our natural and unalienable property. Mental acquirements not only enrich and dignify the possessor; but are productive of the most happy effects to the public in general. Like the common source of light, they shed beams of lustre on all around; and dispel the clouds of darkness, ignorance, and superstition on every side. To the great improvements made in the Arts and Sciences we are indebted for most of the comforts and conveniences of life; and it may be affirmed, without fear of contradiction, that the advancement of no branch of human knowledge has been attended with more beneficial consequences to the community, than the cultivation of the Mathematics. Arithmetic and Geometry are the two grand pillars that support every other branch of science; and to their Prac tical application, together with that of Mechanics, we may atttibute the production of every machine now in use, from the stupendous and powerful steam engine, down to the elegant clock and watch. By the assistance of Arithmetic, we are enabled to make every calculation that business or science requires; and from the principles of Geometry we derive Practical Rules by which we can estimate the areas, solidities, and capaci ties of all kinds of Geometrical Figures. The application of these Rules to finding the areas of figures, is called Mensuration of Superficies; their appli cation to finding the contents of bodies, is called Mensuration of Solids; and when they are applied in finding the capacities of vessels, in ale and wine gallons, and malt bushels, the process is denominated GAUGING. This Art is of such general utility in the common affairs of life, that there are few persons who do not occasionally want its assistance; and to Victuallers, Common Brewers, Maltsters, Distillers, Wine-Merchants, Spirit-Merchants, Soap-Makers, Starch-Makers, and Glass-Makers, it is of considerable moment; for without its aid, they could not. ascertain with accuracy the quantity of those articles which they manufacture. There is, however, another class of persons to whom this Science is indispensably necessary; namely, to those who intend to become Candidates for the Excise; and as so large a portion of His Majesty's Revenue falls under the inspection of the Officers of the Excise, it is certainly of the first importance that they should be made fully ac quainted with the Principles and Practice of Gauging, before they are appointed to discharge so momentous a duty as that with which they are intrusted. Such being the utility of this Science, it is no wonder that so many persons have written on the Subject; but it is certainly matter of surprise, that of all the Treatises which have yet made their appearance, not one of them contains the Theory of Gauging, as it ought to be taught in Schools, united with the Practice, as adopted in the Excise. We have Books of Arithmetic, Mensuration, LandSurveying, Trigonometry, Navigation, Astronomy, Alge bra, Fluxions, and Mechanics, which contain numerous Examples for the Exercise of the Learner; but not one Work on Gauging, that contains more than One Example in each Problem; and that Example wrought out at length. Now, as Schools are the proper places in which the principles of every Science ought to be taught; why not have a Treatise on Gauging adapted for the use of Schools, and at the same time containing every necessary informa tion relating to the Science, as it is practised by the Officers of the Excise? Such are the reasons that have induced the Authors to write the following Work; and whatever may be its merits, it is certainly not a hasty production. It is nearly twenty years since one of its Authors first laid down the Plan, having found the want of such a Book in his avocation of instructing Youth; and both he and his Colleague have employed all their leisure moments for up. wards of five years, in bringing the Plan to maturity. The execution of the Work has cost them much more labour than they at first anticipated; and whoever reads it with attention, and solves all the Questions it contains, will not, they think, charge them with the crime of book making, so much practised in the present day. Having said thus much concerning the advantages of Mathematical Studies, and the motives that led to the production of the Work in question; we will now proceed to give some account of each of the Seven Parts into which it is divided. PART THE FIRST contains a clear and concise view of Vulgar and Decimal Fractions; Square and Cube Roots; with the application of the two latter to the Solution of various Mathematical Problems. A thorough knowledge of these subjects being indispensably necessary in all kinds of measurements; they form the most appropriate Introduction to a Treatise on Practical Gauging. PART THE SECOND contains the description of the Sliding Rule; directions for finding any number upon it; and the application of the different lines to Multiplication, Division, the Rule of Three Direct and Inverse; and the Extraction of the Square and Cube Roots. PART THE THIRD contains such Definitions, Problems, and Theorems in Geometry, as we conceived to be necessary in a Treatise on Practical Gauging. Those who desire to see the subject of Geometry more fully elucidated, are referred to the Elements of Simpson, Emerson, Bonnycastle, Keith, Playfair, and Leslie; to Simson's Euclid, Hutton's Course of Mathematics, and Reynard's Geometria Legitima. The last Work is well adapted to the capacities of Youth; and contains a number of Quæstiones Solvendæ, at the end of each Book; to which an excellent Key has lately been published by the Author. PART THE FOURTH contains the Mensuration of Superficies applied to Gauging. Besides giving Rules for finding the areas of regular and irregular figures; we have treated largely on the method of finding the areas of oval figures, which are not true ellipses, by means of equidistant ordinates. The Honourable Board of Commissioners have, by various General Letters, enjoined the Officers of the Excise to gauge all oval vessels by this method; and it certainly far excels every other, for finding the areas of curvilineal figures, when the nature of the curves cannot be determined. In this Part we have adopted a method entirely different from that of any of our Predecessors; for by considering every plane figure as the base of some vessel whose sides are perpendicular, we have entered immediately on the Practical Part of Gauging, by multiplying the area of the base by the perpendicular depth; and thus we obtain the content of the vessel in ale and wine gallons, and malt bushels. This method has a decided advantage over that |