312 I'KKSKNTMENT AND I'KOOF JN GEOMETRY. 



without obscuring the memory. A good deal of this excellent 

 method may l)e seen in Henrici's most suggestive little work on 

 Congruent Figures. 



(/) Proofs should be Generic and CJenetic rather than 

 Iiujemous. — That is to say, if a hgure or a ])roperty belongs to 

 a family, or may be regarded as having been generated in a 

 special way, proofs should be so chosen as to show forth the 

 relationships of the figure or property, and also, if possible, at 

 the same time to reveal the process of development. 



Besides these maxims, 1 wish to complain that text-book 

 treatment of (leometry is too disjointed. it seems as if some 

 teachers, while rejecting some of Ruclid's best excellences, would 

 have us stick to his antitiuated form. In Euclid's day, when 

 Formal Logic was a new and fashionable game, nothing could 

 be better than his rigid system of cogent syllogisms. Our 

 minds now-a-days do not love to obtrude the skeleton of their 

 processes, and Euclid's method is now to us tasteless and for- 

 bidding. I fear the convenience of having a clear-cut " lesson 

 for to-morrow," or a useful memory-bit for examination ])ur- 

 poses. is deciding in favour of an inferior educational form. 



.Ml tliat I have said hitherto 1 am now going to try to 

 illustrate by a study in the associated circles of the triangle. 

 I may ]M-emise that all the proofs I give are my own, with one 

 exception, and even that excejHion I have so transformed as 

 to give it a new aspect. Not having access to a mathematical 

 librarv. I liave no means of knowing if an\thing I say is really 

 new. 1 merely say that this mode of teaching, and these proofs, 

 have not found their way into any text-book 1 have seen, as it 

 is my belief thev should. The diagrams arc from my own 

 drawings, not ex post facto, but those from wdiich I made the 

 actual studv. I put them forward as illustrating Present- 

 ment. 



Eet me begin with the most fundamental of the associated 

 circles, somewhat quaintly called the Nine-points Circle. And 

 let me give the actual way T introduced it to my junior girl- 

 student class, the members of which hate Geometr)-, and cannot 

 endure to have to prove anything — " What's the use," they say, 

 "when we can see it must be so?" 1 did it in three lessons. 

 The first was a bit of drawing. " Draw a circle, and take 

 anv point inside or out. Rule to or t2 chords through the 

 l)oint. Bisect all the chord-segments. What is the locus of 

 all tho.se mid-points ? " " .\ circle." " How do you know ? " 

 " It looks like it." Then I show them an ellii)se that looks very 

 like a circle, and in one case where there is some inaccuracy in 

 the bisections I show a fourth degree curve more like that locus 

 than a circle. Thus they feel the need of proof, and a little 

 judicious guidance brings us to that proof as indicated in the 

 thick lines of Fig. i. 



