MODERN GEOMETRY: A NEW ELEMENTARY COURSE OF PLANE GEOMETRY. BY DR. R. WORMELL, M.A., WITH AN APPENDIX BY Prescribed by the Board of Education for New Brunswick. London: STREET. 1876. 183. g. Province of New Brunswick, Education Office, Jan. 15th, 1876. The Board of Education, under the authority of the Common Schools Act of 1871, has prescribed this edition of WORMELL'S MODERN GEO. METRY, with an Appendix by Dr. W. BRYDONE JACK, President of the University of New Brunswick, as the only Text Book of PLANE GEOMETRY for use in the Public Schools of this province. THEODORE H, RAND, Chief Superintendent of Education. PREFACE. THE systematic study of the properties of Geometrical figures is now considered 1 an essential branch of ordinary education, for two reasons : 1. Because the demonstration of these properties affords one of the best means of training the mind to habits of thought and accurate reasoning; 2. On account of their industrial and scientific importance. According to current methods of teaching the subject, these two ends are sought by separate and distinct courses; the student of Geometrical demonstrations, on the one hand, rarely having his attention directed to the concrete applications of the science, and, on the other, the student of practical Geometry generally working solely by rule. In the course here laid down, the subject is systematically treated on the assumption that both ends may be attained simultaneously, and with greater ease, rapidity, and efficiency than if pursued separately; the assumption being warranted by the following considerations : 1. We are enabled to avoid the error of giving, without proof, rules for the construction of Geometrical figures, and for the application of arithmetic to their measurements. Rote-teaching and rule-teaching, though they still linger in practice, are now generally condemned in principle as inefficient, since they produce in the pupil an appearance of understanding without the reality, and because isolated rules are easily forgotten, and, even if retained in the memory, are only applicable to the special cases for which they are provided. 2. Because it is neither necessary nor expedient for the efficient progress of the pupil to confine him for any great length of time to the study of Geometry as an art, before introducing him to the science. As soon as a sufficient fund of observations on one kind of simple figure, as, for instance, the angle or triangle, is accumulated, reasoning should begin. In this, as in other departments of knowledge, when the order of nature is followed, experience of things is gained by successive stages, and the reasoning faculties are set to work with every fresh experience. : 3. By reference to the practical application of Geometry for the purpose of explaining and illustrating its terms and statements, the theory is rendered more intelligible and interesting, and much more certain to be remembered, from the associations by which it is thus surrounded. These principles give rise to the leading features of the present work. They were first applied by the author in a somewhat similar work about seven years ago ; but subsequent experience in teaching young pupils, together with the progress lately made in the determination of improved methods of teaching Geometry, as well as the altered requirements of various public examining bodies, have led to the following differences between the present and the former work : 1. The explanations and illustrations of each chapter are made introductory to the strictly logical demonstrations, and are printed in smaller type, to prevent explanation being confounded with definition or illustration being taken for proof. 2. The theorems are numbered consecutively to facilitate reference. 3. In the introductory portions of the several chapters the logical relationship of the propositions is thoroughly examined and different ways of stating certain propositions are given, together with any corollaries or scholiums that may belong to them. 4. Easy exercises and numerical examples are freely introduced. 5. Theorems are separated from problems. 6. The treatment of parallels is postponed till after it has been demonstrated that the exterior angle of a triangle is greater than the interior and opposite angle. 7. Proof is given of the proposition that two sides of a triangle are together greater than the third. 8. The distinction between commensurables and incommensurables is fully exhibited. 9. The nature of ratio and proportion is first explained by the consideration of commensurables, after which the theory is extended by rigorous methods to incommensurables. Also, as from the theory of triangles several tests of equality are deduced, so for proportion several tests are obtained from the definition ; as, for example, the equality of the limits of each ratio, the test of Euclid, and that of the correspondence of the sum of associated magnitudes. A very important feature of the work will be found in the arrangement of the .numerous exercises for solution. These are at first very simple, and increase in difficulty just as rapidly as is needful to concentrate the attention of the pupil at each successive stage. Complete solutions of all these exercises are published separately, and any justification which the position of an exercise may seem to require, will be found in the solution. PREFACE TO THIRD EDITION. For the corrections and additions to this Edition we are indebted to the kindness and patience of Professor Jack, President of the New Brunswick University. Professor Jack has not only carefully and thoroughly revised the text, suggesting the introduction, in their proper places, of several important propositions not included in the previous Editions, but has provided an Appendix, containing a complete analysis of the propositions of the book as compared with those of Euclid. Although this Appendix was prepared especially with a view to the use of the work in the schools of New Brunswick, teachers in England will doubtless find it of considerable use. LONDON, 1876. ADVERTISEMENT. Now Ready, “MODERN PLANE GEOMETRY.” Price 28. 6d. CONTENTS. PAGE PAGE CHAP. I. Lines and Planes. CHAP. III. Circles. Construction - Equality Easy Exercises ............... Abreviations.................. AXIOMS ..................... General-Geometrical - Postulates ............... Triangles ................ , FOR EXERCISE plication of Symmetrical |