PREFACE.

IN presenting to the public a new elementary work on the Principles of Geometry, it can hardly be necessary to defend the having made Euclid's Elements the basis of the work; for while it cannot be denied that many grave faults exist even in the best translations, and that, owing to the advances made in mathematical science since Euclid's day, the demonstrations of many important theorems are wanting in the Elements; it must, on the other hand, be acknowledged that, notwithstanding the numerous attempts which have been made by our best modern geometers to supersede it, the Elements has ever held the chief place in all our universities and colleges.

In the present edition the text of Dr. Simson has been principally followed, but occasionally preference has been given to that of Elrington; the whole has, however, been entirely rewritten, and it is hoped that, in the attempt to render it less verbose, it will not be found that the chain of proof has been in any case weakened. Considerable pains have been taken to distinguish the various parts of the propositions by the adoption of differences in the type; and the references have been grouped in tables under the diagrams, in order to show at sight upon which preceding theorems the truth of each depends.

In the explanatory notes which have been appended it will be found that many additional propositions have been added, and that in several instances other demonstrations have been given.

In the second book it has been endeavoured to point out the relative connection of Geometry and Algebra, and to illustrate by the former the theory of quadratic equations.

In order to remove one of the most practical objections which have been urged against the Elements, namely, its want of methodical arrangement, a classified index has been appended, by

means of which the theorems relating to any particular subject may be immediately found.

In conclusion it must not be omitted to mention the works which have been principally consulted, and to which the present edition must be considered as mainly indebted for any advantages which it may possess. These have been the various editions of the Elements by Simson, Elrington, Tacquet, Barrow, De Chales, Lardner, Potts, Byrne, Playfair, and Thomson, Leslie's Elements of Geometry, Wright's Self-examinations in Euclid, Cresswell's Treatise on Geometry, Bonnycastle's Elements of Geometry, the volume on Geometry in the Library of Useful Knowledge, and a most valuable paper by Professor De Morgan in the Companion to the British Almanac, entitled "Short Supplementary Remarks on the First Six Books of Euclid's Elements."

6, Duke Street, Adelphi,
February 25, 1853.

H. L.

INTRODUCTION.

THE object of Geometry is to investigate and deduce by strict Logic those relations and properties of space and figure which they possess, irrespective of any properties of a physical nature. The whole science of Geometry is based upon certain simple and self-evident truths, from which, by a continuous chain of reasoning, conducted strictly in accordance with the rules of logic, the most important and complicated relations of space and figure are deduced and demonstrated. It is the only science in which hypotheses and theories are unknown, to which experiment and experience have rendered no aid, and whose conclusions are certain and immutable. However much the rules of logic may assist in obtaining true conclusions in the investigations of experimental science, it is only in those of Geometry that its laws are never departed from.

It is therefore evident that the student of Geometry should be perfectly acquainted with formal logic; and we shall give a brief outline of that science before proceeding to the more immediate object of this work. Much discussion has taken place amongst writers on logic as to the scope of the science; some contending that logic includes in its object all the operations of the human understanding necessary to the pursuit of truth; while others would limit it merely to a collection of general rules, by means of which true conclusions may infallibly be derived from true premises. For our present purpose it will suffice to treat the subject in its more limited sense, without entering into the consideration of questions which the first-named writers consider as belonging rather to metaphysics than logic.

The object of reasoning is to extend our knowledge; to enable us, from certain known facts, to derive others of a more general nature; from premises whose truth is evident and acknowledged, to demonstrate the truth of conclusions not in themselves selfevident, and frequently such as, without such proof, would have been regarded as false. In every process of reasoning there are two distinct points to be attended to, namely—

1st. That the propositions employed as premises are not ambiguous, are correctly understood, and are true.

2nd. That the steps by which a conclusion is drawn from those premises are true.

The subject therefore ranges itself properly under two heads; namely, first, an examination into the nature and meaning of propositions, or those premises upon which our reasoning is to be founded; and secondly, an investigation into the mode or form of reasoning to be adopted, that the conclusions drawn may be as true as the premises.

A proposition may be defined to be "An assertion, affirming or denying something;" and, as Mills has justly remarked, "whatever can be an object of belief, or even of disbelief, must, when put into words, assume the form of a proposition;" so that we can never make an assertion, or even hazard a conjecture, without expressing one or more propositions. Now it will be found that the simplest form in which a proposition can exist is the bare statement of the possession of some property, quality, or circumstance, by something. Two objects must be concerned, the something which is the subject of discourse, and the something which is asserted in relation to it; and the proposition is nothing more than the statement of their relative connection. Thus every proposition consists of three parts; namely, 1st, the something of which the assertion is made, termed the subject; 2nd, the sign of affirmation or denial, called the copula; and 3rd, the property, quality, or circumstance asserted, named the predicate. For example, the assertion that "the sun is round" is a proposition of which "the sun is the subject, the verb "is" the copula, and "round" the predicate. Again the exclamation "I think so is a perfect proposition, being equivalent to "such is what I think," in which the word "such" is the subject, and the phrase or sentence "what I think" the predicate. And here it should be observed that the subject and predicate of a proposition may be either simple terms, or names given to objects or their attributes, or they may be complex sentences, themselves containing other propositions. In either case, however, it is essential that the meaning of each should be definite and precise, and perfectly understood, to insure which it is essential that every name or term employed should have but one meaning attached to it, and that that meaning should be perfectly known and understood."

Hobbes has rightly defined a name to be a word taken at pleasure to serve for a mark, which may raise in our mind a thought like to some thought which we had before, and which being pronounced to others, may be to them a sign of what thought the speaker had before in his mind." This definition not only states very precisely what a name is, but shows its use and object, which is simply to suggest to the minds of the speaker and hearer the idea to which it had been attached by the common consent of both. It is therefore evident, that if any word has more than one meaning attached to it, or a meaning unknown to the person who hears it, no certain information can be conveyed to his mind by that word, and it will fail to raise up any certain definite thought or idea; whereas, on the other hand, if that