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and certainty of demonstrable truths :-it is the first condition of success, and the sure means of proficiency in geometrical and in all other studies.

The GRADATIONS IN EUCLID endeavour to carry out the Plan to a greater extent, and with increased distinctness. The Propositions throughout are separated into successive steps; and in the margin, between the vertical lines, direct references are made to the Reasons—the Definitions, Axioms, or preceding Propositions--on which such Construction or Demonstration depends. The method of printing, which has been adopted, also gives a clearer view both of the whole Proposition and of its parts; and familiarises the mind to an orderly and systematic arrangement — so important an auxiliary to all sound progress. By following out a plan of this kind, Learners can scarcely fail to form a distinct conception of what they have to do or to prove, and of the means by which their purpose is to be accomplished.

The EXPLANATORY NOTES direct the Learner's attention to several points of interest connected with the Definitions and Propositions ; and to many of the Propositions is appended an account of the chief PRACTICAL USES to which they may be applied. This is valuable for many reasons, but principally that the Learner may at once see, not simply the theoretical and abstract truths of Geometry, but their direct utility in various ways. There are very many persons who, from studying only the Common Editions of Euclid, which treat exclusively of the Theory of Geometry, never attain to a perception of its importance, and never realise the full advantages of geometrical studies. Indeed, they have read and demonstrated the whole work without perceiving any value in their laborious pursuit, except as an exercise of the memory and mental powers in a very dry course of reasoning. It is the main object of our Gradations in Euclid to combine Theory and Practice, and, as soon as a geometrical truth has been established, to point out its use and application. The Author is thoroughly persuaded that this immediate Combination of Theory and its Application, not only awakens and maintains a livelier interest, but, in fact, leads to a more scholar-like understanding of both, than when they are studied separately, or at wide intervals of time.

The Uses to which the Propositions may be applied are very numerous ; and we have given only a portion of them—more in the way of example, and to point out in some instances the progress of geometrical discovery,

than with the view of exhausting the subject. The various works on Practical Mathematics will supply what may be wanting in this respect. For the full developement of the Uses and Applications of the First and Second Books of Euclid, geometrical principles not worked out in those books must occasionally be introduced; and though it is not strictly logical to employ truths that have not really been established, as the ground-work of further reasoning—now and then, in this part of the work, Lemmas, or truths borrowed from another part of the subject, will be adopted as the foundation of new truths.

The PRACTICAL RESULTS will, it is presumed, be instructive to the Learner in various ways, but especially as exhibiting a synoptical view of all the Problems contained in Euclid's Elements of Plane Geometry. He will not, indeed, have arrived at the means of demonstrating the Problems which lie out of the First and Second Books; but, inasmuch as their construction depends almost entirely on those two books, he will possess, for practical purposes, a knowledge of the methods by which the regular Geometrical Figures, not being sections of a Cone, are to be constructed.

The APPENDIX, containing GEOMETRICAL ANALYSIS and GEOMETRICAL EXERCISES, appeared to the Writer needed for the completion of his plani. e., for comprising a systematic teaching of Geometry, as far as the First and Second Books furnish the means of doing it. The Appendix, and a Key to the Exercises, will each be published separately from the Gradations.

The SKELETON PROPOSITIONS, &c., for pen-and-ink examinations, are arranged and will be published in two Series-one with the references in the margin — the other without those references. The first Series is intended for beginners ; the second, for those who may be reasonably supposed to be prepared for a strict examination. The two Series will be found well adapted to test the Progress of the Learner, and to ascertain how far his knowledge of geometrical principles, and his power to apply them, really extend. The object is, in the first Series, to furnish the Learner, step by step, with the truths from which other truths are to be evolved, but to leave him to work out the results, and from the results, as they arise, to aim at more advanced conclusions : in the second Series, where there are no references supplied 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 in Euclid, 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 printed book aside, reduce his knowledge to a written form, as the references indicate which are given 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 hy previous study of the GRADATIONS in Euclid, 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 Student under examination should also be required to write out the Propositions, &c.,* needed in the Construction and Demonstration, and to supply the * This is required because the repetition of truths and principles gains for

them a more permanent residence in the mind.

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 the 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 Propositions 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 exact. ness ; 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 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 among all classes of the community be rendered more interesting and more practically useful, the objects of the Editor will be accomplished. He desires no worthier calling than to be a fellow-labourer with the many excellent and talented Masters to whom the responsibility is entrusted of training the young in sound learning.

In commendation of the study of Geometry, Principal Hin, of Harvard University, in his First Lessons on the subject, p. 8, declares “Geometry is the most useful of all the Sciences. To understand Geometry will be a great help in lea all other Sciences; and no other Science can be learned unless you know something of Geometry. To study it will make your eye quicker in seeing things, and your hand steadier in doing things. You can draw better, write better, cut out clothes, make boots and shoes, work at any mechanical trade, or learn any art the better for understanding Geometry.”

The principal editions of Euclid to which the Editor is under obligation, are those of Ports 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 Professsor 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.

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