Section 1. Of lines, perpendicular, or inclined, or parallel, to planes 123 2. Of planes which are parallel, or inclined, or perpendicular, to other Section 1. Surfaces and contents of the right cylinder and right cone Section 1. Or great and small circles of the sphere 3. Of equal portions of spherical surface, and the measure of solid angles 194 Part I. Of projection, by lines diverging, and by lines parallel II. Of the plane sections of the right cone, or conic sections ane sections of the oblique cone, of the right cylinder, and of the 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 explanation 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. Book I. Prop. 23 24 Geometry of plane rec 25 tilineal Figures unas 26 sisted by the Theory 27 of Proportion. 29 Definitions 30 Postulates 31 Axioms 32 Prop. 1 33 2 34 3 35 36, omitting Cor. 5 2., Cor.3., 6 and the 7 Scholium 8 37 38 BOOK II. Theory of Proportion, and its application 14, as bereafter to the Geometry of plane rectilineal altered, and figures. omitting the Scholium Definitions 15 Axioms 17 Prop. 1 13 2 19 3 20 4 21 5 22 6 21 [22], omitting Of the Circle. Cor. 2 Definitions [23] Prop. 1 2 1:25 3 [26 4 127 5 (28), omitting the 6 general Scholium 7 Introductory part of 8, omitting the Sect. 3. Schol, 29 11 30 12 31 13 32, omitting 14 Cor. 3 15 34 16 35 17, omitting the 36 Schol. Prop. 18 Prop. 2. Prop. 16 3 17 4 18 5 35 BOOK VI. 7 28 Cor. 2. 8 Book V. Spherical Geometry. 29 10 30 11 Of the Right Cylinder, Definition 12 Prop. 1 13 2 Definitions 3 1. and Cor. 2. 15 Lem. 1 4 5 17 6 7 8 9 5 10 11 12 8 13 24 9 11 Of Lines in different 25 15 Planes, and of Su- 26 11 16 lids contained by 27 12 17 Planes. 28 18 19 Definitious 30 20 Prop. 1 31 21 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 A B let E F be drawn at right angles to CD (12.): EF A shall also be at right angles to AB. For, if A B be not at right angles to E F, through the point E let A' B' be drawn at right angles to EF (post. 5.). Then, by the former part of the proposition, because A' B'. and C 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 E F. 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, B В B F D vii A TABLE OF REFERENCE, 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. Euclid. 1.1. Euclid. IV.3 4 5 6 7 10 11 12 . 63 63 5 7 13 . 5. 5. Cor. 6. 6. Cor. 8 9 10 11 12 13 14 15 15. Cor. 2. 16 17 18 19 20 21 22 23 24 25 26. 16 V. 3 28 and 63 63 II. 2 16 10 11 11. Cor. 1 11. Cur. 3 12 23. Cor. 1 12. Cor. 1 18 17 19 . 9 10 12 13 14 III. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Treatise. 4 7 46 43 44 45 2 2 3 3. Cor. 8. Cor. 1 8 9 9 10 10. Cor. 1 50 47 11 11 5 and 19. Cor. 2 15 15 15 17 48 19 20 20 21 22 and 22. Cor. 1 24 25 27 27 27 27 26 23 57 52 22. Cor.3 36 . . 34 59 III, 55 1 3 3. Cor. 4 6. Cor. 6. Cor. 8. Sch. 8. Sch. 5 7 8. Cor. 1 8. Cor. 1 8. Cor. 1 4. Cor. 4 2 56 2. Cor. 1 2. Cor. 2 14 15 16. Cor. 1 55. Cor. 12 and 14. Cor. 2 12 and 14. Cor. 2 12. Cor. 1 12. Cor. 1 54 15. Cor. 1 and Cor. 2 17 60 61 20 21 21. Cor. 62 10 11 12 13 14 15 16 17 18 19 22 23 24 25 27 28 29 30 31 32 32. Cor. 1 32. Cor. 2. 33 34 21 22 24. Cor. 1 26. Cor. 1 23 20. Cor. 2 14 15 13 . . с D E F 27 . 35 36 37 38 39 40 41 43 13 20. Cor. 1 27 27. Cor. 1 35. Cor. and 39 29 50 31 31 32 33 34 29 30 31 34. Cor. I. 49 32 33 34 35 36 37 IV. 2 5 6 7 8 8. Cor. 9 |