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REV. J. B. LOCK, M.A.,
FELLOW AND BURSAR OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE.
MACMILLAN AND CO.
AND NEW YORK.
[All Rights reserved.]
6-15 4 $ ALE
N the following pages I have attempted to produce an edition of the First Book of Euclid's Elements of Geometry which shall be useful to Teachers.
The existence of a very influential Association for the Improvement of Geometrical Teaching is a sufficient proof that in the opinion of many some of the methods adopted in the past for the teaching Geometry in England have been unsatisfactory.
This Association has I believe done much to point out the defects of our Geometrical teaching, and it has succeeded in inducing the University of Cambridge to announce, that in all her public examinations any proof of Euclid's Propositions which is sound, shall be admitted, provided it does not violate the logical sequence of the propositions as given in Euclid's Elements.
This is a most valuable concession; for it gives Teachers a freedom of which many will gladly avail themselves.
In the following pages I have separated the Theorems from the Problems; and for this reason: A Theorem is a geometrical truth which is based simply on the fundamental ideas and definitions of geometry and is quite independent of any Postulate.
A Problem depends entirely on the Postulates, and the Postulates are practically impossible. We can draw a PICTURE of a straight line, but we cannot draw a straight line.
For example, the truth of the Theorem of Proposition 5 may be proved by the method of Example ii. on page 22,
notwithstanding that we have not yet given a solution of the Problem of how to bisect a given angle. If there is a line which divides the angle ABC into two equal parts, it must cut the line AC in some point E; and then by Prop. 4 the triangles ABE, CBE are equal in all respects.
Similarly we may assume that a triangle DEF, equal in all respects to ABC, can exist notwithstanding that we have not proved a method for drawing a picture of it. If such a triangle can be, then the proof given on page 24 is sound, and independent of Proposition 22.
One advantage of dividing the Theorems from the Problems is this. The figures of the Theorems should be sketched and should be considered only as pictures, while the figures of the Problems should be carefully drawn and should be shewn up as an exercise in Geometrical Drawing. In this way the distinction between theory and practice will be impressed on the beginner.
The examples have been made as easy as possible. They are intended to be worked. If the students cannot do them unaided, the Teacher should do them first, drawing the figure on the Blackboard and explaining them, and then encouraging the student to produce them.
The study of Geometry is not the learning of the Propositions of Euclid by heart. It is the acquiring the power of intelligently using those propositions in the testing and proving other propositions That this remark is not unnecessary I know by my own experience.
It will be observed that I have attempted to give a definition of a straight line. I believe that one of the great difficulties of Euclid as usually taught is that the method of teaching is unscientific. A straight line is defined as that which lies evenly between its extreme parts [why not say a straight line is a finite line which is straight?] and then the real definition of Euclid, which is Two straight lines cannot enclose a space, is called an axiom. Euclid however assumes of a straight line all that I have put into my definition; and, if this is so, is it not much better to say so clearly?
I have also given a definition of an angle.
My own experience is that to tell boys that an angle is the inclination of two lines to one another is just the same as saying (as in the Text-book of the A. I. G. T.) an angle is a pure concept and is incapable of definition. This may possibly be the case; but it greatly staggers the unfortunate beginner to be told so.
I have adopted the term 'straight angle.'
It is I believe quite unnecessary to pretend that two right angles is not an angle. It is an angle; and the sooner it is recognized as such by the learner the sooner do the difficulties of Section V. disappear.
Many Teachers will possibly think that Section I. is misplaced and that this portion of the subject is quite beyond the reach of very young minds. May I suggest that it will be well to take beginners through Section I., without expecting them at once to be able to pass an examination in it? The ideas suggested therein will gradually dawn upon those who study the book intelligently.
The Sections in which the Book is arranged are given in the order in which it seems to me they should be read by a Beginner. But there is no reason why part of Section IV. should not be read before Section III.
A book which deviates so much as this one does from the beaten track must be something of an experiment, and must be capable of great improvement; I venture to ask Teachers who may think it worth improving, to favour me with criticisms and suggestions, so that should it reach a second edition its usefulness may be thereby increased.
J. B. LOCK.