i'Ki:si-:.N"rMEXT a:^I) pRf)OF in gf.omftry. 3 [5 



ABC." "And size?" "Half of ABC?" "Half?" "Well, 

 the dimensions are half." " Now look at ( ) in that triangle: what 

 is it?" '' Whv, it's the orthocentre of that, too." "And are 

 OP. OQ, OR hisected?" "Of course." "Then what do you 

 know of PQR from our last Kindergarten lesson?" "They 

 are on the circum-circle of lA W." " So we have the Nine- 

 points Circle again." " T^ut if it does all that, why do they 

 call it the Nine-])oints?" That. 1 told tliem. is jireciselv what 

 I am asking the mathematicians. 



This dialogue, though of course comi^ressed, is not exag- 

 gerated ; and perhaps it will illustrate my thesis if I sav that 

 these girls, who give such intelligent answers and make such a 

 shrewd comment at the end, are ([tiite likeh' to fail in mathe- 

 matics at the next Matriculation. l)ecause Proof hewilders them 

 and Puzzles annoy them. .My contention is that such correct 

 intuition as they constantly show ought not to be ignored in 

 any educational scheme or test. 



The proof I have just given may l)e shown, ])erliaps more 

 to the satisfaction of a matliematician, l)v invoking another 

 principle. When we draw or imagine a figtire on a plane, we 

 see only one side of that plane. But there is another side to 

 it, and Nature never forgets this. She persists in looking at 

 the " wrong side of the ]jattern " as well as the right. Or, to 

 l^ersonify even more. Nature likes to look at herself in a mirror, 

 where the right hand becomes the left. Hence in (Geometry 

 the mirror is a very useful instrument of Presentment. It 

 reveals symmetry where it has been retained, and restores it 

 where it has l)een Icjst. I hnd. moreover, in teaching, that a 

 mirror with a class of girls has a singtilarly effective power of 

 ri\eting the attention, though they always indignantly deny that 

 it is .so. In doing the last exercise with a mirror, one wants 

 to see throtigh the mirror as well, and in this case it is better 

 to use a bit of plain glass and get one's iiuimIs to look through 

 it at the angle of total reflection, when the whole symmetry 

 will be admirably shown. For drawing jmrposes I shamelessly 

 use tracing paper. Pet not the trained mathematician scoff: 

 a teacher must .gather geometric gear by every wile that's 

 justified by honour. If I may sa^• so. T do not vistialise easily 

 myself, and this des])ised trick has helped me considerably. So 

 v>-e draw a triangle, calling it ( pro])hetically ) DEF ; trace it 

 on the transparent paper ; now turn the triangle round, first on 

 EF. then on ED. then on DE, and mark the new vertices PQR. 

 (Fig. 4.) Rule the new sides, and com])lete the larger triangle 

 ABC. We have now a fresh proof of the Nine-])oints Circle, 

 wliich T need not elaborate. 



