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Thus a child beginning the study of Euclid, while he may contrive to get some notion of the first three propositions of the first book, finds himself hopelessly lost in the mazes of the fourth, fifth, and seventh.

In the second place, Euclid's method of reasoning makes, too large a demand on the powers of a beginner to supply those links in the chain which are understood instead of being expressed. Thus when a child reads : “ Because A is the centre of the circle, AB is equal to AC," his mind fails to supply the missing link in the syllogism, that “all radii of the same circle are equal”; and so he fails to see that the conclusion follows from the premisses.

In the third place, the various editions of Euclid do not help the learner to apply any power of geometrical reasoning he may attain. True, they generally contain deductions to be worked out, but these are given at the end of Euclid's text, and need some intermediate exercises to give the pupil power over them.

Having, as I believed, discovered why Euclid proved so difficult and distasteful, I tried to find a remedy. The problem before me was how to retain Euclid's numbering of the propositions, and Euclid's proofs, and at the same time to make his Elements a more satisfactory book for school use. Gradually I adopted the plan developed in this little book, and found that it acted like a charm. From being the driest and most distasteful lesson of the week, Euclid became one of the pleasantest; difficulties seemed to vanish; and the results exceeded my most sanguine expectations.

I begin, as directed, on page 9 by giving the pupils some acquaintance with geometric terminology, by means of cardboard figures, so that when they meet with such words as angle, triangle, parallel lines, parallelogram, rectilinear, bisection, perpendicular, and so on, they are only meeting with old friends; ncthing gives a child so much confidence in entering on a new subject, as to find he already knows something about it.

I then introduce them to a simple syllogism as on page 10; then to a geometric syllogism, then to a chain of such syllogisms in a simple proposition, quite within the powers of any child beginning the subject. In the earlier problems of Euclid brought before their notice, the syllogisms are first written out in full; and the pupil discovers that each proof is merely a chain—longer or shorter-of simple syllogisms, such as he can readily understand. The procf is then given in contracted syllogisms as nearly as possible in the words of Euclid.

In the next place the book is divided into three parts. Part I. treats only of the Problems of Euclid as far as proposition 23. This is done by assuming as axioms, for the present, Euclid's fourth and eighth propositions. The problems, being much easier than the theorems, and for the most part shorter, at the same time admit of more elaborate treatment, and the pupils become thoroughly accustomed to the language and method of Euclid, before meeting with any difficult proposition.

In Part II., the theorems on the subject of angles and triangles are dealt with. Euclid's order is departed from, but Euclid's numbering is preserved, the object being to give the theorems as nearly as possible in the order of difficulty. This, at the same time, admits of a more systematic arrangement than Euclid gives. For instance, all the propositions treating of the equality of triangles are grouped together. In this part also the extended syllogisms are not given; but explanatory passages are inserted and assistance given where experience has shown them to be needed.

BEING AN INTRODUCTION TO THE STUDY

OF THE

FIRST BOOK OF EUCLID'S ELEMENTS.

BY

T. S. TAYLOR,

AUTHOR OF

THE “FIRST PRINCIPLES” SERIES OF EDUCATIONAL WORKS.

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LONDON: RELFE BROTHERS,
6, CHARTERHOUSE BUILDINGS, ALDERSGATE.

183. q. 120

BY THE SAME AUTHOR.

FIRST PRINCIPLES OF ENGLISH HISTORY.

Fcap. 8vo., 1s. cloth. " What does it matter to nine-tenths of mankind whether William married Blanche or Matilda- whether this king had three wives or six-whether that one was crowned at Westminster or Winchester ? But it does concern each and all of us to know why a devastating war raged between the king and his parliament: what principles were involved ; and how it might have been avoided : to trace the growth of trial by jury through successive generations ; to learn how one man's character influenced the whole realm for good or evil; to know that our deeds outlive us and bear fruit of which we little dream."-Author's Preface.

FIRST PRINCIPLES OF FRENCH HISTORY.

Fcap. 8vo., 1s. cloth. “Mr. Taylor's experience has given him an insight into the tastes and powers of children, and has enabled him to write quite the best Elementary History of France which has yet come under our notice,"

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FIRST PRINCIPLES OF ROMAN HISTORY.

Fcap. 8vo., 1s. cloth. Intended to give clear ideas as to the leading facts of Roman History and to show the successive steps in the growth and decline of the Roman Empire and people."

FIRST PRINCIPLES OF ENGLISH GRAMMAR.

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FIRST PRINCIPLES OF MODERN HISTORY.

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