Elements of Geometry, Theoretical and Practical: Including Construction by the Right Line and by the Circle Together with the Mensuration of All the Elementary Plane Figures and Solids

From the Preface. THE present work has been written under the influence of opinions in which the author cannot be regarded as singular. He has presumed that a modern systematical treatise on the Elements of Geometry is essential to modern scientific education, and that any attempt to present those elements with the view of facilitating their more effectual acquirement would meet the approbation of mathematical instructors. In an elementary course of exact science, the principal requisites are conceived to be a natural sequence of its different parts, and a suitable degree of brevity in their exposition. These have been constantly kept in view in the Treatise now offered to the public; and their observance must greatly contribute to the improvement of students in Geometry, and to the removal of difficulties which have hitherto retarded their progress. With respect to the importance of arrangement, and its effects in the study of this science, the author has had ample experience. He has long observed, that in proportion as the different subjects occupying the attention of students succeeded in natural or in artificial connexion, their ideas appeared distinct or confused, and the impressions left in their minds were of permanent or of momentary continuance. From these and other considerations relative to demonstrative and inductive science, the author premised in the composition of this work, the following as fundamental principles in regard to the manner and order in which the objects of synthetic Geometry should be introduced and treated. First, that the possible existence and characteristic property of every object considered, should be established previously to forming its definition. Secondly, that every object defined should be regarded as establishing the existence of an equal object. Thirdly, that two or more independent objects should be separately and fully treated; and that the several properties of each should succeed in regular order as to the relative position, proportion, and extension of its corresponding parts. Fourthly, that different connected objects should follow in natural sequence from the simple to the more complex; and that a like arrangement should be observed in treating two or more objects viewed in connexion. The necessity of admitting these principles must be obvious to every reader conversant with Geometry, and the advantages which result from them will be easily imagined. Superfluous exertions of memory will undoubtedly be avoided; forced and artificial views will be obviated; an accurate and comprehensive knowledge of the subject will be readily acquired; and the reasoning and inventive powers of the student greatly improved and properly directed. That the present treatise will to the full extent realize the advantages here enumerated, the author has been induced to hope.