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London: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MARIA LANE.
Glasgow: 50, WELLINGTON STREET.

Leipzig: F. A. BROCKHAUS.
New York: THE MACMILLAN COMPANY.
Bombay and Calcutta: MACMILLAN AND CO., LTD.

[All Rights reserved.)

PRACTICAL AND THEORETICAL

BY

C. GODFREY, M.A.
SENIOR MATHEMATICAL MASTER AT WINCHESTER COLLEGE,

AND

A. W. SIDDONS, M.A.
FELLOW OF JESUS COLLEGE, CAMBRIDGE; ASSISTANT MASTER

AT HARROW SCHOOL.

CAMBRIDGE
AT THE UNIVERSITY PRESS

1903

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PREFACE.

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|HE aim of the authors of the present work has been to

produce a book which will help to make Geometry an attractive subject to the average British boy or girl.

The new schedule of geometry recently adopted by Cambridge has been taken as a basis of operations. These regulations will affect candidates for the Previous Examination after March 1904.

It has been found easy to follow this schedule closely and at the same time to have regard to the reformed schedules of various other examinations, such as. Oxford and Cambridge Locals, Oxford Responsions, together with the examinations of the University of London, and the Civil Service Commissioners. The reports of the British Association and of the Mathematical Association have been very helpful.

The book opens with a course of experimental work; great pains have been taken to make the exercises perfectly explicit and free from ambiguity. The beginner is taught to use instruments, to measure accurately lines and angles (this will in future be regarded as an indispensable part of geometrical work), to construct and recognize the simpler plane and solid figures, to solve problems by drawing to scale. At the same time he is led to discover many geometrical truths which are proved later; he should be encouraged to put into words and make notes of any such discoveries. There is much in this part which will be useful revision work for more advanced pupils.

3

Then follows the course of Theoretical Geometry, which is divided into four "books.' The experimental method is still prominent, in the shape of exercises leading up to propositions.

The sequence of theorems is Euclidean in form, but greatly simplified by the omission of non-essentials, and by the use of hypothetical constructions. There is reason to hope that it is now possible to adopt a sequence (not differing very greatly from that of Euclid) which will be generally accepted for some time to come.

The treatment of problems is practical, though proofs are given; for this part of the subject the present work is designed to fulfil the purposes of a book on geometrical drawing.

Among the exercises, some are experimental and lead up to future propositions, some are graphical and numerical illustrations of known propositions, some are riders' of the ordinary type*. In a great number of the earlier exercises the figures are given. There is a collection of exercises on plotting loci and envelopes; a subject which is found interesting, and introduces the learner to other curves than the circle and straight line.

Book I. deals with the subject matter of Euclid I. 1—34; angles at a point, parallels, angles of polygons, the triangle, the parallelogram, sub-division of straight lines, the earliest constructions and loci.

Book · II. treats of area. The notion of area is enforced by a large number of exercises to be worked on squared paper, the use of coordinates being explained incidentally. Euclid's second book appears in a new garb as geometrical illustrations of algebraical identities.

* We are indebted to the kindness of Mr R. Levett and of Messrs Swan Sonnenschein and Co. for permission to use a few of the riders from The Elements of Plane Geometry issued under the auspices of the A. I.G. T.

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