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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.
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 CD are, each of them at right angles to E F, A'B' is parallel to CD. 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 EF 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,
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.