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PLANE AND SOLID GEOMETRY.
H. W. WATSON, M.A.
Sometime Fellow of Trinity College, Cambridge;
LONGMANS, GREEN, AND CO.
THE study of Elementary Geometry (at least in England) has been for a long time identified with one particular treatise, accepted as a standard. At the present moment there is a wide-spread dissatisfaction with that treatise; but there is very little agreement as to the manner in which it may be best improved. The most suitable Preface, therefore, to a new work on Geometry would appear to consist in an enumeration of the main features of its agreement or disagreement with Euclid, and an attempt as far as possible to justify such agreement or disagreement in each particular instance.
I. The Work agrees with Euclid in retaining the syllogistic form throughout. Many objections, strong and ably urged, have been alleged against this method of treatment by modern writers. It is said that the study of Geometry for its own sake is thereby made subordinate to its study as a logical discipline; and that the detailed syllogistic form into which all the demonstrations are thrown is a source of obscurity to beginners, and damaging to true geometrical freedom and power.
Now, whatever truth there may be in these charges as applied to Euclid's treatise, they do not appear to be applicable to the syllogistic form of statement in itself. No doubt there is very much unnecessary prolixity in some of Euclid's demonstrations. He has in many cases perversely refused to draw the most general inference possible from his premises, even where by doing so he would not have lengthened or embarrassed his reasoning, and the result has been much useless repetition, and vain expenditure of words. As one illustration out of very many which might be adduced of the truth of this assertion the 26th Proposition of Book I. may be instanced. The demonstration of this proposition would have been in no degree lengthened if Euclid had extended it to the equality of the triangles in all respects. As it is, he has restricted himself to the sides and angles, and accordingly, when in Proposition 34 he has proved the equality of the opposite sides and angles of a parallelogram by means of this Proposition 26, he is obliged to recur to Proposition 4, with its long and cumbrous enunciation, to arrive at the equality of the areas of the two triangles into which the diameter divides the parallelogram.
Nothing can be said for a waste of time and words like this, but it would be unfair to charge the syllogistic method generally with such an obvious defect in its application. The truth seems to be that the syllogistic method, if properly applied, may be made