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The present work provides a course of Elementary Geometry based on the recommendations of the Mathematical Association and on the schedule recently proposed and adopted at Cambridge.
The principles which governed these proposals have been confirmed by the issue of revised schedules for all the more important Examinations, and they are now so generally accepted by teachers that they need no discussion here. It is enough to note the following points:
(i) We agree that a pupil should gain his first geometrical ideas from a short preliminary course of a practical and experimental character. A suitable introduction to the present book would consist of Easy Exercises in Drawing to illustrate the subject matter of the Definitions; Measurements of Lines and Angles; Use of Compasses and Protractor; Problems on Bisection, Perpendiculars, and Parallels; Use of Set Squares; The Construction of Triangles and Quadrilaterals. These -problems should be accompanied by informal explanation, and the results verified by measurement. Concurrently, there should be a series of exercises in Drawing and Measurement designed to lead inductively to the more important Theorems of Part I. [Euc. I. 1-34].* While strongly advocating some such introductory lessons, we may point out that our book, as far as it goes, is complete in itself, and from the first is illustrated by numerical and graphical examples of the easiest types. Thus, throughout the whole work, a graphical and experimental course is provided side by side with the usual deductive exercises.
(ii) Theorems and Problems are arranged in separate but parallel courses, intended to be studied pari passu. This arrangement is made possible by the use, now generally sanctioned, of Hypothetical Constructions. These, before being employed in the text, are carefully specified, and referred to the Axioms on which they depend.
* Such an introductory course is now furnished by our Lessons in Experimental and Practical Geometry.
H.S.o. I.-IV. i
(iii) The subject is placed on the basis of Commensurable Magnitudes. By this means, certain difficulties which are wholly beyond the grasp of a young learner are postponed, and a wide field of graphical and numerical illustration is opened. Moreover the fundamental Theorems on Areas (hardly less than those on Proportion) may thus be reduced in number, greatly simplified, and brought into line with practical applications.
(iv) An attempt has been made to curtail the excessive body of text which the demands of Examinations have hitherto forced as "bookwork" on a beginner's memory. Even of the Theorems here given a certain number (which we have distinguished with an asterisk) might be omitted or postponed at the discretion of the teacher. And the formal propositions for which—as such—teacher and pupil are held responsible, might perhaps be' still further limited to those which make the landmarks of Elementary Geometry. Time so gained should be used in getting the pupil to apply his knowledge; and the working of examples should be made as important a part of a lesson in Geometry as it is so considered in Arithmetic and Algebra.
Though we have not always followed Euclid's order of Propositions, we think it desirable for the present, in. regard to the subject-matter of Euclid Book I. to preserve the essentials of his logical sequence. Our departure from Euclid's treatment of Areas has already been mentioned; the only other important divergence in this section of the work is the position of I. 26 (Theorem 17), which we place after I. 32 (Theorem 16), thus getting rid of the tedious and uninstructive Second Case. In subsequent Parts a freer treatment in respect of logical order has been followed.
As regards the presentment of the propositions, we have constantly kept in mind the needs of that large class of students, who, without special aptitude for mathematical study, and under no necessity for acquiring technical knowledge, may and do derive real intellectual advantage from lessons in pure deductive reasoning. Nothing has as yet been devised as effective for this purpose as the Euclidean form of proof; and in our opinion no excuse is needed for treating the earlier proofs with that fulness which we have always found necessary in our experience as teachers.
The examples are numerous and for the most part easy. They have been very carefully arranged, and are distributed throughout the text in immediate connection with the propositions on which they depend. A special feature is the large number of examples involving graphical or numerical work. The answers to these have been printed on perforated pages, so that they may easily be removed if it is found that access to numerical results is a source of temptation in examples involving measurement.
We are indebted to several friends for advice and suggestions. In particular we wish to express our thanks to Mr. H. C. Payne and Mr. H. C. Beaven of Clifton College for the valuable assistance they have rendered in reading the proof sheets and checking the answers to some of the numerical exercises.
H. S. HALL.
PREFATORY NOTE TO THE SECOND EDITION.
In the present edition some further steps have been taken towards the curtailment of bookwork by reducing certain less important propositions (e.g. Euclid I. 22, 43, 44) to the rank of exercises. Room has thus been found for more numerical and graphical exercises, and experimental work such as that leading to the Theorem of Pythagoras.
Theorem 22 (page 62), in the shape recommended in the Cambridge Schedule, replaces the equivalent proposition given as Additional Theorem A (page 60) in previous editions.
In the case of a few problems (e.g. Problems 23, 28, 29) it has been thought more instructive to justify the construction by a preliminary analysis than by the usual formal proof.
H. S. HALL.