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The Sequence of Theorems in School Geometry*
Published online by Cambridge University Press: 03 November 2016
Extract
The subject of my address has recently been “ventilated” in more than one organ of educational opinion; and it is stated that a committee will shortly be assembled to explore the possibility of an escape from the present chaos to the sweet simplicity of an agreed and authoritative sequence.
I know nothing about the constitution of the committee nor about the proposals that are likely to be brought before it; nor had it been announced or foreshadowed when I suggested the subject for discussion to-night. My intention in suggesting that subject was merely to revive and develop further certain proposals brought forward in an address given to the Association at a time when many of its members were engaged in a vastly more serious discussion elsewhere. I venture, however, to hope that the circumstances I have referred to may add materially to the usefulness of our debate.
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- Copyright © Mathematical Association 1922
Footnotes
*
The substance of a lecture to the Bristol Branch of the Mathematical Association, 17th March, 1922. Some replies on points raised in the discussion have been incorporated.
References
page note 65 † At the Annual Meeting of January, 1917.
page note 66 * I do not, of course, deny that the logical coherence of a geometrical system fortifies our belief in the truth of all its parts; my point is merely that this result of a logical inquiry into geometry is not the main reason why we undertake it. To avoid another possible source of misunderstanding, I add that I deliberately ignore here, as too abstract for the school-boy, the standpoint of the truly “pure” geometer. The geometry I have in view is the scientific study of actual space.
page note 67 * The Foundations of Geometry. A translation is published by the Open Court Publishing Company.
page note 67 † He might equally well have started with the assumption that if AB=A′B′, LA=LA′, and LB=LB′, then AC=A′C′ and BC=B′C′. From this it follows that LC=LC′, and Euclid I. 4 can also be deduced.
page note 69 * The difficulty with regard to maps is, indeed, to persuade him that they are not merely reduced diagrams of the areas they represent.
page note 69 † The work should include the enlargement and reduction of drawings and a simple treatment of perspective and should incidentally teach the correct technical use of the term “similar.” Mr. Fawdry pointed out in the discussion that boys will call all ellipses (for instance) “similar.” This, I suggest, is because, according to the ordinary meaning of the word, they are similar—just as all triangles are. A new technical term, without misleading associations, would be acceptable. Would homomorphic or identiform be too alarming?
page note 69 ‡ Some of the Universities now permit candidates for matriculation to refer proofs to the principle of similarity. This is a great advance. It tends to give similarity the place here claimed for it, and it enables teachers to substitute for Euclid’s cumbrous proofs of I. 47, III. 35, 36, the simple arguments recommended long ago in Mr. W. C. Fletcher’s text-books—from which many of us, in our early days, learnt a great deal. One must also refer gratefully to the help given to the cause by authorities in the service of the Board of Education whose position compels them to be anonymous.
page note 69 § I concur with Mr. Carson’s opinion (Mathematical Education, p. 104) that everyone who proceeds to a University “should gain some slight idea of the nature of non-Euclidean geometry,” and I submit that what follows here is not a bad introduction to the subject.
page note 71 * This would, for example, be the case ii the first pair were 3.463 … and 3.482 … while the second pair were 3.471 … and 3.502 …, but not if the second pair were 3.483 … and 3.496 ….
page note 72 * It wil be of interest to quote his actual words: “Praesumo tandem … ut communem notionem
Datae unicunque Figurae, Similem aliam cujuscunque magnitudinis possibilem esse.
Hoc enim (propter quantitates continuas in infinitum divisibiles, pariter atque in infinitum augibiies) videtur ipsa Quantitatis natura fluere; figuram scilicet quamlibet continue posse (retenta figurae specie) tam minui, tam augeri in infinitum.” From Opera (1693), vol. ii. p. 674. I have followed up the references in Bonola’s invaluable Non-Euclidean Geometry (Open Court Series).