1 PREFACE, N the following Treatise are laid down nitude, with such other Parts of the Ma. tbematicks only, as are necessarily leading thereto, and their Application in the Investigating and Demonstrating the several Theorems for. measuring those plain Surfaces and Solids, which more immediately belong to Gauging : The whole being originally design’d for the benefit of the Officers of the Excise, that they might not only.. be acquainted with the most easy and exact Me thods for Practice, but be at the same time cere tain of the Truth of the Rules made use of. Care has been taken to avoid meddling with the Methods practised by fome others; or any way to be concerned in refuting their Errors; but here the only business is, to give the neceffary Rules for the just and ready Practice of the Art of Gauging ; fbewing the Manner of com ming at those Rules, from Principles within themfelves the most perfeet of all human Knowledge. For that End, it is hoped, every thing bere treated of, is explain'd in fo easy a manner, as to A be be within the reach even of Beginners, that Tho the Practice of Gauging may be al- And as no Writer hitherto has thought fit The first, as an Introduction to the other In the fecond Part, there are some of the primary Principles of plain Geometry, with those of the Conic Sections, as preparatory to what follows : Where such general Propositions, for the Measuring of Plane Surfaces and som tids, are laid down, that the Rules for most of the common.ones, are deduced by way of Corollaries. You have bere particular Rules for Menfuration all right-lined Plane Surfaces, Ciro cles, Conic Sections, and for Solids, such as Parala lelopipedons; Prisms, Conoids, Spindles, with the Segments and Fruflums of these : Some of these perhaps were not published before, such I fuppole; are the Series for the Circle (in Pag. 100.). for the Hyperbola (in Pag. 109.) and the Theos rems for measuring the Gircular, Elliptic, and Hyperbolic Spindles ( in Pag. 156.) Then are given the Investigation of Theorems, for reducing these Solids nearly into Cylinders of the fame Length: From whence is deduced, accafionally, the common fixed Multipliers, and fivemen bow far they may be depended upon, in the Practice of Gauging. Next, a general Theorem is given, for the Mensuration of all Pyramidical or Conic Hoofs, and the same is accomplisk'd in the three Hoofs of an upright Cone, and that of the Square Pyramid : On this depends the method of Gauging an inclined Fyramidical Conical Tun. Then the Method is shewn of Measuring by Approximation, which, if brought into Uje, A 2 would would render the Art of Gauging far more conpleat, than it has been bitherto: For by this Method, the Measure of any Cafk, Tun, Cops per, or Still may be found within any desired Degree of Truth, and tkat by only taking a sufficient Number of Dimensions, wbich will ever be less than what is required in the common methods of gauging Coppers, and the Conclufons will be much nearer the Truth. But for Calks, there is, besides the usual Data, required only a Diameter in the Middle between the Head and the Bung; from which the Form of the Cask, in some Drgree, is determined. This perhaps may at first be look'd upon as a Design to introduce Novelty in a Matter fufficiently confirm’d by Experience, and by that means be rejested as a speculative Nicety. But if it be considered that the present Practice of Gauging depends upon assigning fome known Form to the Tun or Cask to be gauged 3 and without assuming that Form, we cannot inake one Step towards computing its Meafure : So that if we are not exact in our Af fumption, it is not possible we should be exa£t in its Content. And it may be with truth afirm'd, that there never was, strictly speaking, à Calk in the Shape of any one of those Solid Figures whose Names they bear ; hince it never could happen so, unless perfe&tly by Accident : The Maker baving no thougkt of designing them as such; and besides, skould a Cask approach nearly |