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NOV 18 1931
ENTERED, according to Act of Congress, in the year
STEREOTYPED BY RICHARD H. HOBBS,
TAERE are two methods of investigating the principles of Geometry. The only method known to the ancients was independent of the aid of Algebra. This method has been so completely developed by Euclid, as to leave little room for improvement. It is true, modern writers have arrived at many of his conclusions by more simple and concise methods; but, in so doing, they have, in most instances, sacrificed the rigor of logical demonstration, which so justly constitutes the great merit of his writings.
While but little room is left for improving on the model of Euclid, the modern geometer, by bringing to his aid the principles of Algebra, has greatly enriched the geometry of the ancients, by the discovery of many beautiful relations of magnitudes, which probably would never have been brought to light by the old method.
In this work, which is after the model of Euclid, we have not strictly copied any one author, but have endeavored to select from all the sources within our reach, such parts as we deemed best adapted to our wants. In the solid geometry, or geometry of three dimensions, we have made free use of Peter Barlow's arrangement, as given in the Encyclopædia Metropolitana ; which, indeed, is but a slight modification of LEGENDRE'S method.
We have found, from experience in teaching, that, as a general thing, beginners in the study of geometry consider it as a dry, uninteresting science. They have but little difficulty in following the 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, 1847.
ELEMENTS OF GEOMETRY,
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.
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.) From any point to another point an infinite number of lines may be drawn, but only one straight line can be drawn; all the others will have flexure in a greater or less degree. The straight line 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