where there are no references in the margin, the object is to make the examinations strict and thorough, yet so as to be conducted on one uniform plan. This uniformity will be found greatly to assist Examiners, when they compare the examination papers together for the purpose of deciding on their respective merits. The Skeleton Propositions may be used either simultaneously with the Gradations, or, after the first and second books have been read in any of the usual editions of SIMSON'S Euclid, as a recapitulation of the ground already gone over: if used simultaneously, the Learner must first study the Definitions and Propositions in their order, and then, laying the Gradations aside, reduce his knowledge to a written form, as the references indicate in the vertical columns of the Skeleton Propositions; but if used as a recapitulatory exercise, a course in some respects different is recommended. In the recapitulatory exercise, the following plan is recommended for adoption :—first, that the Learner should give in writing a statement of the meaning of various Geometrical Terms, of the nature of Geometrical Reasoning, and of the application of Algebra and Arithmetic to Geometry; secondly, that he should fill in,--not by copying from any book, but from the stores of his own mind and thought, trained by previous study of the GRADATIONS or of some similar work,-the Definitions, Postulates, and Axioms of which the leading words are printed; and thirdly, that he should proceed to take the Propositions in order, and write out the proofs at large, as the printed forms and references in the margin indicate this should be done systematically in all the propositions, beginning with those truths already established which are required for the Construction and Demonstration, and then taking in order the Exposition, the Data and Quæsita, or the Hypothesis and Conclusion, the Construction with its methods, and the Demonstration with its proofs separated from each other, and given, step by step, in regular progression. : For the thorough Examinations, that series of the Skeleton Propositions must be used which contains the General Enunciation only, without any references printed in the margin. The Spaces for the exposition, construction, and demonstration, are retained, and also the vertical lines within which the Learners themselves are to place the references; but this is done simply for the purpose of securing a uniformity of plan in the written examinations, and for the convenience of Examiners. The Stu dent under examination should be required to write out the propositions, &c.,* needed in the Construction and Demonstration, and to supply the references to the various geometrical truths by which the steps of the proposition are established. No figures, or diagrams, are given in either of the Series of Skeleton Propositions; as it is more conducive to the Learner's sound progress that he be left entirely to himself to construct these. The Uses and Applications of the Propositions, at least in a brief way, --and where requisite, the Algebraical and Arithmetical Illustrations,— should not be neglected: it is in these that the practical advantage of abstract truths is rendered apparent. It is imperative that the Teacher should revise each Proposition after it has been written out, and note all misapprehensions and inaccuracies before the Learner proceeds to the following proposition. In Self-Tuition the Learner must consult the Gradations, and by them correct the already filled-up Skeleton; but he must be faithful to himself, and to his own improvement, by not consulting the Gradations as a Key, until he has first worked out and written down his own conception of what the demonstration demands. He will thus build up for himself and of himself; he will make the dead bones of the Skeleton live, clothe them with flesh and sinews, and round them off in all their proper proportions. A course of this kind followed faithfully through two books of the Elements of Geometry, will scarcely fail to render the Student competent by himself to master the other books of Euclid; and, should he desire it, by the same means. He will have learned the value of method and exactness; and expert in these, he will attain a solid and durable knowledge of Geometrical Principles. At the present day nearly every edition of Euclid's Elements must be, more or less, a compilation, in which the Author draws freely on the labours of his predecessors. "The Gradations" are, in a great degree, of this character; and an open acknowledgment will suffice, once for all, to *This is required on the principle that the repetition of old truths gains for repel any charge of intentionally claiming what belongs to others. It is affectation to pretend to great originality on a subject which has, like Geometry, for so many centuries exercised men's minds. If by the methods employed in the following pages, the Study of Geometry be rendered more interesting and more practically useful,—and especially if his work be adapted to the wants of that numerous class of Learners, the Pupils in Parochial and similar Schools,—the objects of the Editor will be accomplished. He desires no worthier calling than to be a fellowlabourer with the many excellent and talented Masters to whom, in the National, British, and other Public Schools of the United Kingdom, the responsibility is entrusted of training the young in sound learning. The principal editions of Euclid to which the Editor is under obligation, are those of POTTS and LARDNER, and of an old Writer who professes to give “the uses of each Proposition in all the parts of the Mathematicks.” An Exemplification and recommendation of the plan pursued in the Gradations and in the Skeleton Propositions, may be found in the preface to LARDNER'S Euclid, and in a Treatise on the Study and Difficulties of Mathematics, p. 74, attributed to Professor DE MORGAN. He is also indebted to various persons, Schoolmasters and others, for valuable suggestions, which he takes this opportunity to acknowledge. The work is longer than the Author at first contemplated; but he trusts that the additions,—especially the Practical Results, and the Exercises,―will add considerably to its usefulness and value. INTRODUCTION. SECTION I. GRADUAL GROWTH OF GEOMETRY AND OF THE ELEMENTS OF EUCLID. GEOMETRY has been defined in general terms to be,-"the Science of Space." It investigates the properties of lines, surfaces, and solids, and the relations which exist between them. Plane Geometry investigates the properties of space under the two aspects of length and breadth; Solid Geometry, under the three, of length, breadth, and thickness. It is the consideration of the Elements of Plane Geometry on which we are about to enter. In Geometry, land-measuring, as the word denotes (from gè, earth or land, and metron, a measure), was in its origin an Art, and not a Science: it embraced probably a system of rules, more or less complete, for performing the simpler operations of land-surveying; but these rules rested on no regularly demonstrated principles, they were the offspring rather of experiment and individual skill, than of scientific research. the same way poems-even some of the noblest-were composed before the principles of poetry had been collected into a system; languages were spoken, long before a grammar had been compiled; and men reasoned and debated before they possessed either a logic or a rhetoric: so measurements were made, while as yet there was no accurate theory of measuring,- -no abstract speculations concerning space and its properties. The points and lines of such a Geometry were necessarily visible quantities. A mark, which men could see, would be their point; a measuring rod, or string, which they could handle, their line; a wall, or a hedge, or a mound of earth, their boundary. The first advance beyond this towards an B abstract Geometry, would be to identify the instruments which they used in measuring, with the lines and boundaries themselves; the finger's breadth, or the cubit, the foot, or the pace, would become representatives of a certain length without reference to the shape. It was only as the ideas and perceptions of those who cultivated the art of measuring grew more refined and subtile, that a Geometry would be evolved, such as Mathematicians understand by the term, in which a point marks only position; a line, extension from point to point; and surface, a space enclosed by mathematical lines. The Truths of Geometry, as a science, regularly as they are laid down and deduced in the Elements of Euclid, were not worked out by one mind, nor established in any systematic order. Some were discovered in one age, some, in another; two or three propositions by one philosopher, and two or three, by some one else. The collection of geometrical truths had thus a gradual growth, until it received completion at the hands of Euclid of Alexandria. Thales, who predicted the eclipse that happened в. c. 609, is said to have brought Geometry from Egypt, and to have established by demonstration Propositions 5, 15, and 26, of Bk. i.; 31, iii.; and 2, 3, 4, and 5, of bk. iv. Pythagoras, born about 570 B. C., was the first who gave to Geometry a scientific form, and discovered Propositions 32 and 47 of bk. i.: Oenopides, a follower of Pythagoras, added the 12th and 23rd of bk.i.: and Eudoxas, B. C. 366, a friend of Plato, wrote the doctrine of proportion as developed in the fifth book of the Elements. These assertions may not rest on the firmest authority, yet they shew, even if they are only surmises, that Geometry was regarded by the Greeks as a science of very gradual formation, receiving accessions from age to age, and from various countries. It was at first a set of rules, until philosophy investigated the principles on which the rules were founded, and out of the chaos created knowledge. According to Proclus, EUCLID of Alexandria flourished in the reign of the first Ptolemy, B. c. 323–283. To him belongs the glory, for such it is, of having collected into a well-arranged system, the scattered principles and truths of Geometry, and of having produced a work, which, after standing the test of above twenty centuries, seems destined to remain the Standard Geometry for ages to come. |