demonstration, and arriving at a full conviction of its truth; but they ask, What if the proposition is true? What use can be made of it? Now, to meet these difficulties, we have all along in the body of the work added, in a smaller sized type, such remarks, suggestions and practical applications as we have found from experience to interest the pupil. Our object has not been to multiply these practical applications, but merely to give in their proper places a few of the more simple cases, such as would naturally suggest themselves to the mind of a successful teacher. A few examples, given in this way, will excite in the pupil a desire to invent for himself still further applications, thus keeping up a lively interest in the study of this most important branch of education. The arrangement of the work is such as to make the text, which is given in the larger sized type, wholly independent of the explanatory matter in small type. The course is, therefore, complete with the omission of the practical portion. In an appendix, we have given the solution of a few geometrical problems by the aid of algebra; thus showing the facility with which many difficult cases are made to yield, under the influence of the analytical method of investigation. We have also taken this opportunity to exhibit some beautiful and interesting theorems, by translating the results of algebraical deductions into the language of geometry. GEORGE R. PERKINS. UTICA, September, 1947. ELEMENTS OF GEOMETRY, WITH PRACTICAL APPLICATIONS. FIRST BOOK. THE PRINCIPLES. GEOMETRY is the science of extension. It considers the extent of distance, extent of surface, and the extent of capacity or solid content. The name geometry is derived from two Greek words, signifying land and to measure. (Art. 1.) Egypt is supposed to have been the birthplace of this beautiful and exact science, where the annual inundations of the Nile rendered it of peculiar value to the inhabitants as a means of ascertaining their effaced boundaries. At the present time it embraces the measurement of the earth and of the heavens. Its principles are applicable to magnitudes of all kinds. There is scarcely any mechanical art which does not receive great assistance from Geometry. DEFINITIONS OF MAGNITUDES. I. A solid or body is a magnitude having three dimensions : length, breadth, and thickness. A II. A surface is the limit or boundary of a solid, having two dimensions : length, and breadth. III. A line is the limit or boundary of a surface, having only one dimension : length. IV. A point is not a magnitude. It has no dimension in any direction, but simply position. Hence, the extremities of lines are points. Also, the place of intersection of two lines is a point. (2.) The common notion of a point is derived from the extremity of some slender body, such as the end of a common sewing needle. This being perceptible to the senses, is a physical point, and not a mathematical point; for, by the definition, a point has no magnitude. V. A straight line is the shortest distance between two points. (3.) Among the infinite number of lines which can be imagined, having different degrees of flexure, one only corresponds with the straight line, namely, the one which has no flexure. The outlines of the different objects of nature are, in general, presented to us in the form of curved lines, some of which are very graceful and pleasing to the eye. (4.) In accordance with the above definition, if a fine flexible string be stretched between its two extremities, it will assume, nearly, the direction of a straight line. Owing to the weight of the string, it will necessarily be bent downwards. If, however, we could suppose the string devoid of weight, it would then produce a straight physical line, which will approach more nearly to the mathematical line as the size of the string is diminished. (5.) All the lines which we form upon paper or upon the blackboard, for the purpose of illustrating the principles of Geometry, are physical lines. Indeed, it is impossible to form a mathematical line, but we may, however, conceive of such lines, and this we must always do in our geometrical reasoning ; and for the want of a better method, we use the physical lines as representatives of the mathematical lines which we wish to consider. (6.) In ornamental gardening, the sides of walks, the rows of plants, shrubs and trees, etc. are determined by stretching a flexible cord between their extremities. In carpentry and other arts, straight lines are formed upon plane surfaces by stretching upon the surface a flexible cord, previously rubbed over with chalk. The middle portion of the cord is then raised, and allowed to recoil by its elasticity, thus leaving upon the surface a chalked line. (7.) Another definition of a straight line is as follows : When a line is such, that the eye being placed near one extremity so as to cause it to conceal the other extremity, it shall, at the same time, hide from view all other portions of the line; then such line is called a straight line. This definition is due to Plato. A practical application of this definition is used by artisans, in bringing the eye to range along the direction of the line under consideration, technically called sighting. VI. Every line which is not a straight line is called a curved line. When we hereafter speak of a line, unless otherwise expressed, we shall mean a straight line. Thus, AB and CD are straight lines ; GH and KL are ; curved lines. The extremities of these lines, as well as their intersections F and M, are points. VII. A plane surface, or simply a plane, is a surface, in which, if two points be taken at pleasure, and connected by a straight line, that line will be wholly in the surface. a (8.) A practical test, in accordance with this definition, is employed by artisans to determine whether a surface is plane. They take a rod or rule whose edge is straight, and apply it in various directions upon the surface under consideration, observing whether the edge of the rule, or as it is technically called the straighte-dge, coincides in all positions with the surface; if so, the surface is a plane. (9.) The practical miller, when he wishes to dress his millstones to a plane surface, rubs the straight-edge with paint, and then applies it in various directions upon the face of the stones; thus showing, by the transfer of the paint, which are the highest portions of the stone. These portions are dressed down; and the process again repeated, until the face of the stone has been brought as near a plane surface as may be deemed necessary. The action of the carpenter's plane is founded upon the same principle. VIII. Every surface which is not plane, is called a curved surface. (10.) The plane surface may be regarded as one particular kind of surface out of the infinite varieties which can be imagined. To the eye, many of the curved surfaces are far more graceful and pleasing than the plane. IX. When two straight lines AB, C AC, meet each other, the space included between the lines is called an angle. The point of intersection A, is the vertex of the angle ; and Α. the lines AB, AC are the sides of B the angle. Perhaps it would be better to define an angle as the opening between two lines which meet. An angle is sometimes referred to by simply naming the letter at its vertex, as the angle A ; but usually by naming the three letters, as the angle BAC, or CAB, observing to place the letter, at the vertex, in the middle. |