PREFACE. HAVING, in the former parts of this series, treated of the elements of geometry, trigonometry, conic sections, &c., it now remains for us, in accordance with our original design, to make such application of the former principles, as to elicit such other truths or principles as depend on their various combinations, and to investigate the relations of such subjects as pertain to the higher geometry. And, without attempting to give a full and perfect treatise on the subject, which would require volumes, we shall endeavor to present some portions of this ancient subject in a new dress; hoping, thereby, to render its beauties more plainly visible, and its oracles more intelligible. Some new solids are introduced into this volume, the most important of which is a class termed revoloids; which, from their organization, seem to serve as a connecting link between rectilinear and curvelinear solids. The properties of those solids are discussed, and their surfaces and solidities are determined. Some new curves are also introduced and investigated, among which is the revoloidal curve, whose quadrature is determined; and, from its relation to the circle, and also to rectilinear figures, we are enabled to approximate to the circle's quadrature to an indefinite extent. During the investigation of this subject, other important properties of the circle will be developed, by which the area of the segment of circle whose arc and sine are known, may be computed with as little labor as that of the area of a triangle whose base and perpendicular are given. We have also introduced into this work a mode of con struction for variable quantities, or such magnitudes as depend on variable factors; and have adapted a notation, embracing some of the principles of the calculus, by which variable magnitudes may be algebraically discussed, and their conditions rendered intelligible. By this notation, some of the more difficult geometrical subjects are susceptible of the most elegant solution; and we are also enabled to get a definite algebraic expression for the circle's quadrature, in terms of the diameter. The mensuration of such superficies and solids as depend on the higher geometry, follows at the close of the work. From the hasty manner with which a considerable portion of this work has been prepared, it can hardly be presumed to be entirely free from errors; but it is believed that if any errors exist, they are such as involve no important principle. The author, with these remarks, submits the work to the consideration of an intelligent public. CONTENTS. BOOK I. On the Species and Quadrature of the Sections of Elementary Solids, embracing the Parabola, Ellipse, and Hyperbola, Definitions, The Sections of a Cone, The Sections of a Polyedroid, The Sections of a Prism, &c., Quadrature of the Parabola, On the Ellipse, its Quadrature, &c., BOOK II. Solid Sections, or Segments of Solids of Revolution, Ungulas, &c., Definitions, Cylindrical Ungulas and Segments, Conical Segments and Ungulas, Segments and Ungulas of an Elliptical Cylinder, Spherical Ungulas and Segments, Parabolic Prisms and Ungulas, BOOK III. On Revoloids and Solids formed by the Revolution of the Conic Sections, Quadrature of the Surface of a Revoloid determined, Revolvoids and Ungulas equivalent to a Sphere or Spheroid, Segments of Revolvoid, Parabolic Revoloid, its Cubature determined, Sections of Solids formed by the Revolution of the Conic Sections, BOOK IV. PAGE. 7 7 8 10 11 16 20 23 31 34 On the Revolvoidal Curve, the Rectification of the Ellipse, and other Curves, and on the Quadrature of the Circle, Rectification of the Revoloidal Curve, On the Circle's Circumference, and its Quadrature, Curve of the Circle's Quadrature, BOOK V. On the Production and Resolution of Geometrical Magnitudes, Production of Surfaces and Solids, CHAP. II.-On the Construction of Quantities whose Elements are a series either of constant or variable quantities, Explanation of Principles, and Notation, Construction of Variables, Construction of Curves from their Equations, Quadrature of the Circle expressed algebraically, in terms of known functions of the diameter, 157 157 159 159 160 . 161 . 166 166 169 173 173 174 Equivalent Constructions for the Equations of Curvelinear CHAP. III.-The Differential and Integral Calculus, Differential Calculus, Integral Calculus, CHAP. IV. On the Centre of Surfaces and Solids, the Virtual Centre The Virtual Centre of a Circular Arc, The Virtual Centre of the Surface of a Solid, CHAP. V.-On the Relations of Lines, Surfaces, and Solids, gene rated by the motion of the Virtual Centre, General Proposition, MENSURATION OF SUPERFICIES. Circular Segments and Zones, To find the Circumference of a Circle, or any Arc, Mensuration of the Ellipse, Mensuration of the Parabola, Mensuration of the Hyperbola, . To find the Area of any Plane Surface by Equi-distant Ordinates, Magnitude of Bodies, by their Weight and Specific Gravity, QUESTIONS FOR EXERCISE, Description of an Instrument for Measuring Distances and Heights by 212 . 214 . 216 . 218 . 219 221 . 224 227 228 . 233 234 236 242 . 248 |