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CHIEFLY FROM

THE TEXTS OF SIMSON AND PLAYFAIR,

WITH CORRECTIONS.

AN IMPROVED FIFTH BOOK ;
A SUPPLEMENT, ON INCOMMENSURABLE QUANTITIES,
AND AN APPENDIX, ON PLANE TRIGONOMETRY; TOGETHER WITH

NOTES AND COMMENTS.

DESIGNED FOR

THE JUNIOR MATHEMATICAL CLASS IN BELFAST COLLEGE.

BY J. R. YOUNG,
PROFESSOR OF MATHEMATICS IN BELFAST COLLEGE.

LONDON:
JOHN SOUTER, 131, FLEET STREET.

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PREFACE.

In this edition of the Elements the first four books are principally from the texts of Simson and Playfair, corrected wherever the reasoning appeared to me to be inconclusive, and modified wherever any considerable brevity or simplicity seemed likely to result from the change : but no departure from the words of these excellent geometers has been made without careful deliberation, nor without some degree of hesitation, except in cases where a regard to geometrical accuracy rendered it imperative. Such cases, however, are certainly more numerous than many persons suspect; and therefore more alterations of the established text occur in this than in any of the preceding editions. It would be tedious formally to enumerate these: I have therefore, in most instances, introduced the necessary emendations, as they seemed to be demanded, without note or comment.

The experienced geometer will easily discover where these changes occur, and how far they deserve the designation I have ventured to give them.

The Fifth Book is a total departure from Euclid's. It is designed to lessen, if not entirely to remove, those obstacles which students have often found insurmountable in the doctrine of geometrical proportion. No mention is made of ratio, the geometrical abstraction which is the root of most of the difficulties and perplexities felt by learners in the study of Euclid's Fifth Book. It appears to me that the term is the exclusive property of arithmetic, and represents no geometrical idea. This Fifth

Book I have, for many years, constantly employed in teaching, in place of that of Euclid, and always with success.

The critical remarks and brief comments which run through this edition will, I hope, enlarge the student's acquaintance with the subject, and incite him to consult other sources of information. The “ Additional Notes” at the end are few and brief; though much instructive matter might, no doubt, have been introduced under this head. As, however, I have already given, in my octavo “Geometry," a copious account of the discussions into which the perplexities of parallel lines and proportion have led geometers; and have entered at length upon several topics of interest to the critical reader, I have thought it better to refer at once to that treatise, for this kind of miscellaneous information, rather than to increase the bulk and price of this volume by transcribing it here.

The Supplement on “Incommensurable Quantities" is an addition to the “ Elements,” necessary to the completion of the subject of Proportion. The neglect of this class of quantities in the common editions of Euclid, has long been considered by geometers as a capital defect.

The Appendix contains a brief, but, I hope, a sufficiently perspicuous treatise on “ Plane Trigonometry.” It may serve to prepare the student for a more easy entrance into the analytical theory of the subject, when he arrives at that stage of his mathematical progress. Or, in the event of his terminating his scientific course with this volume, the practical information which the tract conveys may be turned to useful account in the professional avocations of active life.

J. R. YOUNG. Belfast College ; Oct. 1, 1838.

THE

ELEMENTS OF GEOMETRY.

INTRODUCTION.

Our notions of a point, a line, and a surface, with the definitions of which Euclid commences, are derived from our ideas of extension in general, and precede all definition; and the object of Euclid, in defining these abstractions, seems less to convey the conceptions of the things defined than to express those conceptions with due precision.

Our ideas of extension are derived primarily from the external world; but we can contemplate extension in the abstract, or apart from the body extended, just as we can think of motion apart from the body moved.

The properties of extension and its boundaries form the subject matter of GeoMETRY; which is, therefore, conversant with form--figure-bulk. We cannot give a sensible representation of these, but in connexion with body or material substance; yet by an act of abstraction we may have a conception of form, and make it alone the subject of our reasonings.

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