ELEMENTARY COURSE. IN compiling the present Treatise, the leading object has been to furnish the "Library In the above table the propositions only are mentioned: when corollaries or scholia are attached to any of the propositions, they are likewise to be attended to, unless the contrary is expressly stated. The sections of Problems (omitting III. 64, Case 4, the solution of which depends on a lemma of the scholium following II. 38.) will, it is apprehended, be found rather entertaining and serviceable to a beginner than otherwise; they are not necessary, however, and are therefore omit- The demonstration of the converse part of Book I. Prop. 14., is attended with a difficulty which is stated at some length in page 11, as we have been anxious that the student should be fully aware of its existence. It will be better, however, in a first perusal, to avoid this difficulty by making, at once, the following assumption : "Through the same point there cannot pass two different straight lines, each of The converse part of Prop. 14, viz. that "parallel straight lines are at right angles to the same straight line," will then be demonstrated as follows:- Let A B be parallel to CD, and from any point E of For, if A B be not at right angles to E F, through the It will be found that the Course just laid down, excepting the sixth Book of it Showing the Propositions and Corollaries of Simson's Euclid which are to be found in the present Treatise, and the parts of the Treatise corresponding to them. GEOMETRY. BOOK I. $1. Definitions-§ 2. First Theorems § 3. Parallels-§ 4. Parallelograms -§ 5. Rectangles under the parts of divided Lines-§ 6. Relations of the Sides of Triangles-§ 7. Problems. SECTION 1. Definitions. GEOMETRY is the science of extension. The subjects which it considers are extent of distance, extent of surface, and extent of capacity or solid content. The name Geometry is derived from two Greek words, signifying land and to measure. Hence it would appear that the measurement of land was the most important (perhaps the only) use to which this science was, in the first instance, applied. Egypt is described to have been its birth-place, where the annual inundations of the Nile rendered it of peculiar value to the inhabitants as a means of ascertaining their effaced boundaries. From the Egyptians the ancient Greeks derived their acquaintance with it; and, in the hands of this acute people, it was carried, from a state of comparative nothingness, to a degree of perfection which has scarcely been advanced by succeeding ages. If, how ever, as a science, Geometry has made but little progress, since it was so successfully cultivated by the Greeks, its uses have been both multiplied and extended. In the present day it embraces the measurement equally of the earth and of the heavens: it forms with arithmetic the basis of all accurate conclusions in the mixed sciences: and there is scarcely any mechanical art, our views of which may not be improved by an acquaintance with it. The truths of Geometry are founded upon definitions, each furnishing at once an exact notion of the thing defined, and the groundwork of all conclusions relating to it. The leading definitions are as follows: 1. A solid is a magnitude having three dimensions-length, breadth, and thick ness. 2. A surface is the boundary of a solid, having length and breadth only. 3. A line is the boundary of a surface, having length only. 4. A point is the extremity of a line, ther length, nor breadth, nor thickness. having no dimensions of any kind-nei 5. (Euc. i. def. 4.)* A right line, or straight line, is that which lies evenly between its extreme points. When the word "line" is used by itself in the following pages, a straight line is to be understood. 6. Any line of which no part is a right line is called a curve. If a curve be cut by a straight line in two points, the curve is said to be concave towards that side upon which the straight line lies, and towards the other side, con vex. 7. (Euc. i. def. 7.) A plane surface, or plane, is that, in which any two points whatsoever being taken, the straight line between them lies wholly in that surface. 8. A surface, of which no part is plane, is said to be curved. 9. If there be two straight lines in the same plane, which meet one another in a point, they are said to form at that point a plane rectilineal angle. A B The magnitude of an angle does not depend upon the length of its legs, that is, of the straight lines by which it is contained, but upon the opening between them, or the extent to which they are separated the one from the other. Thus, the angle B A C is greater than the angle BA D, by the angle D A C. If there be only one angle at the point A, it may be denoted by the letter A alone, as "the angle A;" but if there be more angles at the same point, it becomes necessary to indicate the containing sides of each, in order to distinguish it from the *This and the like references are to Simson's Euclid, the Roman numeral indicating the book, and the other the proposition. When the reference is to a definition, as in the present instance, or to an axiom, the same is particularized by the initial syllable Def. or Ax. The most important definitions only, which are taken from Euclid, and stated in nearly the same words, are here referred to, B |