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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 proof 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.
The Third Part treats of the doctrine of parallels, and an effort has been made to simplify for the pupil the great stumbling-block of this part of the Elements, the twelfth axiom.
An important feature of the book will be found in the exercises to be worked by the pupil, one or more of which is attached to each proposition; and hints are given as to the method of working, by which it is believed a foundation will be laid for the successful working of deductions in examinations and elsewhere.
Among other details, the data and quæsita of each proposition are placed clearly before the mind of the pupil—lengthy enunciations being given in tabular form. Nothing is more common than for a child to flounder through the proof of a theorem with only the vaguest notion of what he is trying to prove. Thick black type has been employed to bring the more important parts of each problem clearly before the eyes ; and in the diagrams, the method has been adopted of showing in one figure simply the data of the proposition, and then giving the lines of construction in subsequent figures.
I believe that nothing tends more to make children hate Euclid - and it is an admitted fact that the majority of children do hate Euclid-than to find the first steps they take in it hopelessly difficult; and what is more hopelessly difficult for an average child beginning the study than Euclid's fifth proposition ? By the plan adopted in this work, the pupil is not asked to attack that proposition till he approaches the end of the second part.
T. S. TAYLOR.
FIRST PRINCIPLES OF EUCLID.
BEING AN INTRODUCTION TO THE STUDY OF THE
FIRST BOOK OF EUCLID'S ELEMENTS.
Before commencing these Lessons the pupil should have acquired an intimate acquaintance with the various forms of ordinary Geometrical Figures ; with straight and curved lines; with angles of different magnitudes. · They should know what is meant by perpendicular and parallel lines, bisection of lines and angles, and the construction of circles from radii, etc. This knowledge should be acquired from viva voce teaching ; the pupils being supplied with pieces of cardboard or wood to represent lines, circles, and surfaces, and being required to construct with them the various figures, and to make actual perpendiculars, parallels, bisections, etc.
A SYLLOGISM. Look at these two statements :
(a.) Every boy has a head.
(6.) John is a boy. What third statement can we make, drawn entirely from these two ? Of course it is :
(c.) John has a head. And
you see at once, that if (a) and (6) are true, (c) must be true also.
Let us look at the three statements together.
Every boy has a head.
John is a boy. Therefore, John has a head.
This arrangement of three statements, in which the third must follow (or be true) if the first and second be true, is called a Syllogism, or Argument.
The first two statements are called Premisses ; and the third is called the Conclusion.
Now suppose we put the letters X, Y, and Z, for the words boy, head, and John, our syllogism will read thus :
Every X has Y.
Z is X. Therefore Z has Y. This is a sort of skeleton syllogism, having the general form in which every syllogism must be expressed. And we may put any words for the letters X, Y, Z as long as the first and second statements thus made are true. Thus :—Every vertebrate animal has a back-bone,
The horse is a vertebrate animal.
... The horse has a back-bone. Here we have put vertebrate animal for X, back-bone for Y, and horse for Z.
(The three dots .. stand for therefore.) Caution.—The pupil must not suppose that a correct conclusion can be drawn from every pair of statements he may meet with, even though the statements themselves be true. In forming syllogisms he must be careful to have them strictly in the form of the skeleton syllogism.
EXERCISE.—Construct five syllogisms.