REMARKS ON GEOMETRICAL REASONING, AND ON ARITHMETIC AND ALGEBRA APPLIED TO GEOMETRY ; PRACTICAL RESULTS AND EXERCISES: BY HENRY GREEN, A.M. "GEOMETRY IS, PERHAPS, OF ALL THE PARTS OF MATHEMATICS, THAT WHICH OUGHT TO BE --Lacroix, p. 306. MANCHESTER : JOHN HEYWOOD, 170, DEANSGATE. LONDON : EDINBURGI: BELL & BRADFUTE. GLASGOW: W. HAMILTON. 18 5 8 783. c./2. . PREFACE, RESPECTING THE GRADATIONS AND SKELETON PROPOSITIONS, ETC. The chief aim of the Author or Compiler of the Gradations in Euclid, with Skeleton Propositions, &c., for Written Examinations, has been to furnish a useful book to those in the humbler stations of life, who attend our Parochial and similar Schools, who have not much time for the pursuit of Geometrical Studies, and to whom, therefore, the Practical Application of whatever they learn is of great importance. He is, however, persuaded that those who have both time and full opportunity, either in Public Schools or in Colleges, for attaining proficiency in the Higher Mathematics, will find an Introduction, such as is given in this work, very suitable to prepare them more thoroughly to appreciate Geometrical Truths, and to take an interest in them as the ground-work of accurate science. The INTRODUCTION is of general use to all Students of Geometry : it contains—A brief account of the Gradual Growth of Geometry and of the Elements of Euclid ;—The Signs and Contractions that may be employed : and some Remarks-on the Nature of Geometrical Reasoning, -on the Application of Arithmetic and Algebra to Geometry, -on Incommensurable Quantities,—and on Written and Oral Examinations. The subjects treated of pre-suppose, indeed, that the Learner has a clear understanding of Fractions, common and decimal,—of the extraction of the Square Root, -and of the introductory principles of Algebra : but this knowledge is indispensable for those who would really master the Elements of Geometry. The Editions Euclid by Potts and BLAKELOCK, have shown the advantages of printing separately and distinctly the parts of a Proposition and of its Demonstration : it is a plan which undoubtedly gives very valuable help to Learners in attaining a more exact acquaintance with the Principles on which Geometry, as a science, is founded. No argument is here needed to prove the importance of being able to estimate the force 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 they depend. 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 PRACTICAL USES to which the proposition may be applied. This is valuable for many reasons,—but chiefly 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 realize the full advantages of geometrical studies. It is the main object of the 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. Only a portion of the Uses to which the Propositions may be applied has been given,-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 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 already 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 synoptic view of all the Problems contained in the Elements of Plane Geometry. He will not, indeed, have arrived at the means of demonstrating the prob. lems, 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 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 plan, i. 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, |