ELEMENTARY COURSE. In compiling the present Treatise, the leading object has been to furnish the “Library of Useful Knowledge” with a body of geometrical information, in which nothing might be wanting that seemed necessary or desirable, whether to the correct explapation and solid establishing of the science, or to its application in the different branches of natural philosophy. Such an object, it is plain, can never be accomplished by a mere elementary course, which has solely in view the instruction of beginners: it implies many discussions and distinctions, many theorems, scholia, and even whole sections of matter, which it is better that a beginner should pass by, while he confines his attention to the few and simple but important propositions to which perpetual reference is made, and which may be regarded as constituting the high road of Geometry. At the same time, the purposes of instruction have not been lost sight of; and accordingly, while the present work may be considered sufficiently extensive to answer every useful purpose, it will be found also to include an elementary course of study complete in itself, by the help of which a person totally unacquainted with the subject may become his own instructor, and advance by easy steps to a competent knowledge of it. With this view, the beginner has only to confine himself to the following portions of the entire work. Prop. 23 40 42 43 44 Cor. Book I. Prop. 7 37 24 Geometry of plane rec Prop. 38, omitting the 25 [9] tilineal Figures unas Lemmas in 26 sisted by the Theory the Schol, [10] 27 Til 39 of Proportion. 29 †12] Definitions 30 41 Postulates 31 1141 Axioms 32 115 Prop. 1 33 1161 50, omitting the f18 719 Book III. and the [21] 7 Scholium [22], omitting Of the Circle. Cor. 2 Definitions [23] Prop. 1 10, omitting the 24 2 3 T26 4 Theory of Proportion, 271 [28], omitting the 7 general Scholium Introductory part of 8, omitting the omitting the Sect. 3. Schol. 29 11 Axioms 30 12 31 13 2 14 Cor, 3 15 20 34 16 21 35 17, omitting the 22 36 Schol, 15 vi Prop. 18 EL ELEMENTARY COURSE. Prop. 16 3 17 4 18 5 35 Book VI. 7 28 Cor. 2. Book V. 8 Spherical Geometry. 29 10 30 11 Of the Right Cylinder, Definition 31 12 Right Cone, and Prop. 13 2 Definitions 3 1. and Cor. 2. 15 Lem. 1 4 16 5 17 6 7 8 9 5 10 11 12 8 13 24 9 14 Of Lines in different 25 10 15 Planes, and of So- 26 11 16 lids contained by 27 12 17 Planes. 28 18 19 Definitions 30 20 21 Prop. 15 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 omitted in the table. 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 which is parallel to the same straight line." 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 B' point E let A' B' be drawn at right angles to EF (post. 5.), D Then, by the former part of the proposition, because A' B' and Č D are, each of them at right angles to EF, A' B' is parallel to C D. But A B is parallel to CD. Therefore, through the same point E there pass the two straight lines A B and A'B', each of which'is parallel to CD. But it is assumed that this is impossible. Therefore, the supposition that A B is not at right angles to E F is impossible ; that is, A B is at right angles to EF. It will be found that the Course just laid down, excepting the sixth Book of it only, is not of much greater extent, nor very different in point of matter from that of Euclid, whose “ Elements” have at all times been justly esteemed a model not only of easy and progressive instruction in Geometry, but of accuracy and perspicuity in reasoning. A perusal of this work, as translated and edited by Simson, though certainly not essential to an acquaintance with geometry, is strongly recommended to the student, E A B с E 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. . 4 5 5. Cor, 6 6. Cor. 8 9 10 11 12 13 . Euclid. 2 14 2 . . . 26 . . 63 14 15 15. Cor. 2. 16 17 18 19 20 21 22 23 24 25 26 Treatise. 4 7 Cor. 2 19 20 Cor. 1 Treatise. 30. Cor. 58 1 2. Cor. 2 Cor. 2 Cor. 2 Euclid. Treatise. IV. 3 III. 62 4 59 5 59 6 63 7 63 10 28 11 63 12. 27. Cor. 1 and 63 13 26 63 16 10 11. Cor. 1 10 11. Cor. 3 11 12 12 23. Cor. 1 13 12. Cor. 1 14 18 2 21 24. Cor. 1 23 26. Cor. 1 24 25 25 20. Cor. 2 A 14 17 19 22 . . 35 36 37 38 39 40 41 43 44 46 46. Cor. 47 48 . 27 13 20. Cor. 1 27 27. Cor. 1 35. Cor. and 39 29 50 31 31 32 33 34 34. Cor. I. 49 36 62 GEOMETRY. ver. B BOOK I. 3. A line is the boundary of a surface, § 1. Definitions---$ 2. First Theorems, having length only. § 3. Parallels—4. Parallelograms 4. A point is the extremity of a line, 7.9.5. Rectangles under the parts of having no dimensions of any kind-neidivided Lines— 6. Relations of the ther length, nor breadth, nor thickness. Sides of Triangles—$ 7. Problems. 5. (Euc. i. def. 4.)* A right line, or Section 1. Definitions. straight line, is that which lies evenly between its extreme points. GEOMETRY is the science of extension. When the word “line" is used by itself The subjects which it considers are ex- in the following pages, a straight line is tent of distance, extent of surface, and to be understood. extent of capacity or solid content. 6. Any line of which no part is The name Geometry is derived from a right line is called a curve. two Greek words, signifying land and to If a curve be cut by a straight measure. Hence it would appear that line in two points, the curve is said the measurement of land was the most to be concave towards that side important (perhaps the only) use to upon which the straight line lies, which this science was, in the first in- and towards the other side, constance, applied. Egypt is described to have been its birth-place, where the an 7. (Euc. i. def. 7.) A plane surface, nual inundations of the Nile rendered or plane, is that, in which any two points it of peculiar value to the inhabitants as whatsoever being taken, the straight line a means of ascertaining their effaced between them lies wholly in that surface. boundaries. From the Egyptians the 8. A surface, of which no part is plane, ancient Greeks derived their acquaint- is said to be curved. ance with it; and, in the hands of this 9. If there be two acute people, it was carried, from a state straight lines in the of comparative nothingness, to a degree same plane, which meet of perfection which has scarcely been one another in a point, advanced by succeeding ages. If, how they are said to form at ever, as a science, Geometry has made that point a plane rectibut little progress, since it was so suc lineal angle. cessfully cultivated by the Greeks, its The magnitude of an angle does not uses have been both multiplied and ex- depend upon the length of its legs, that tended. In the present day it embraces is, of the straight lines by which it is the measurement equally of the earth contained, but upon the opening between and of the heavens: it forms with arith them, or the extent to which they are metic the basis of all accurate conclu- separated the one from the other. Thus, sions in the mixed sciences: and there the angle B A C is greater than the anis scarcely any mechanical art, our views gle B AD, by the angle D A C. of which may not be improved by an If there be only one angle at the point acquaintance with it. A, it may be denoted by the letter A alone, The truths of Geometry are founded as “the angle A ;" but if there be more upon definitions, each furnishing at once angles at the same point, it becomes nean exact notion of the thing defined, and cessary to indicate the containing sides of the groundwork of all conclusions re- each, in order to distinguish it from the lating to it. The leading definitions are as follows: # This and the like references are to Simson's 1. A solid is a magnitude having three Euclid, the Roman numeral indicating the book, and dimensions-length, breadth, and thick 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 syl. 2. A surface is the boundary of a lable Def. or Ax. The most important definitions only, which are taken from Euclid, and stated in solid, having length and breadth only. nearly the same words, are here referred to. a 2 A. ness. B |