sesses some advantage. But a few changes have been introduced into the second and third books from those of Legendre; a few useful propositions have, however, been add In the fourth book, by discussing the subject of Proportion and proportional figures before that of the squares of the sides of right angled triangles, we are enabled to present a new demonstration for that proposition founded on the proportionality of the sides of similar triangles, at once elegant and satisfactory, and to deduce directly from this, in the most conclusive manner, the general property that similar triangles and polygons are proportional to the squares on their homologous sides, &c. Some curious propositions follow toward the close of this book, some of which are original, and some have been selected from other authors, with such modification as was deemed necessary to correspond with the style of this work. In the fifth book, also, will be found much new matter. It is unnecessary to particularize. The sixth book consists of the Isoperimetry of plane figures, and is succeeded by Notes illustrating certain portons of the subject to which they refer. The application of the principles embraced in elementary plane geometry, to Mensuration, closes this part of the se ries. The second part consists of elementary solid geometry based also on Legendre's Elements, but is considerably ex ded beyond the ordinary limits of similar works. Some new solids are introduced, and it is believed that part relating to solids of revolution, especially of the surface and solidity of a sphere, is more rigorous and satisfactory than in any other work. The third part consists of spherical geometry, analytical trigonometry, the application of trigonometry to the mensuration of heights and distances, and trigonometrical surveying. &c. The application of algebra to geometry; Conic Sections, are also embraced in this part. The fourth part treats of such solids of revolution as depend on the higher geometry, and the solid sections or segments of such as are embraced in the elementary part. In this volume some new solids are introduced. By the discassions in this volume, we are enabled to arrive at most of the useful properties of such solids as are bounded by curve surfaces, including the segments of solids of revolution, by the simple elementary principles of geometry, some of which have only been heretofore obtained through the medium of Luxions, or the calculus. The quadrature of the circle, and the rectification of the elliptical circumference and some other curves, are also discussed. Some new properties of the circle, and other curves are developed, enabling us to pursue our investigations in relation to them, by pure geometry. We are also enabled to show that, although it may not be possible for us to express the circumference of a circle numerically in terms of its diameter, yet that it may be expressed in terms of some known function of that diameter, and hence that it is possible to get an algebraic expression for its value in known quantities. The series closes with the application of the former principles to the mensuration of such figures, whether plane or solid, as depend on the higher geometry. Some rules are obtained, much shorter and easier than any others in use, for the determination of some of the more difficult problems. CONTENTS. THE APPLICATION OF NUMBERS TO MAGNITUDES-RATIO AND PROPORTION-DEF1- NITION OF TERMS-EXPLANATION OF SYMBOLS-AXIOMS-POSTULATES-DEFINI- TIONS AND EXPLANATIONS OF PRINCIPLES USED IN PROPORTION-GENERAL DIS- THE CIRCLE AND MEASUREMENT OF ANGLES-DEFINITIONS-GENERAL DISCUSSISON AND PROPOSITIONS-PROBLEMS RELATING TO THE SECOND AND THIRD BOOKS. BOOK FIFTH. THE MENSURATION OF SUCH PLANES AS DEPEND ON ELEMENTARY PRINCIPLES- ELEMENTS OF GEOMETRY. BOOK I. GENERAL PRINCIPLES. GEOMETRY is the science which treats of the measurement and comparison of magnitude, and the relations of locality. Magnitude can have but three dimensions; length, breadth and thickness; all of which are necessary to constitute a body, or solid. It is important, however, to consider magnitude under three distinct denominations; of lines, surfaces and solids and thus the science of Geometry becomes divided into three principal branches: the first part treating of lines described upon the same plane, and of the surfaces which they enclose; the second, of lines situated in different planes, and of the relations of those planes to each other; and the third part contemplating body under its several dimensions of length, breadth and thickness. Lines are obviously the boundaries of surfaces, and surfaces are the boundaries of solids; it is equally obvious that a line being mere length, without either breadth or thickness, can exist only as the boundary of a surface, and that a surface, being absolutely without thickness, can exist only as an attribute of body. Although, therefore, it cannot be supposed that a line, or a surface, can have separate or independent existence, the fact will not in the smallest degree interrupt or embarrass our reasonings, in considering these several attributes of body or space, each apart from the others, nothing more being requisite than the abstracting these others from our inquiry; so that in considering lines, length only is recognized, and in contemplating surfaces, length and breadth are combined and thickness excluded. Geometry is necessarily dependent on numbers for the de |