I'RKSENTMENT AND PROOF TN CKOM FTR^'. 3II 



the mind for the period of proof. Pupils should not be be- 

 wildered by a double simultaneous unfamiliarity. Woodwork 

 ought to do much of this for boys; girls are usually without 

 this advantage. 



(b) Presentment is often useful zcithout Proof. — In phy- 

 sical science we do not prove everything : we tind it sufficient 

 to train our pupils in the modes of proving, but when an investi- 

 gation is beyond their reach, we give them results which others 

 have proved. Why not also in Geometry? For example. I 

 always let mv class of jiuiiors draw tangents to a circle with 

 the ruler alone, because it is the most accurate way : hereafter 

 they will better appreciate the theory of polars when they come 

 to it. See also a clever specimen of such presentment in 

 Professor Boys' charming little book on Soap Bubbles, where 

 he shows how to get the Conic Sections by the shadows of a 

 candlestick. At Matriculation stage, pupils may well be already 

 drawing ellipses and parabolas in various ways, and even tracing 

 simple forms of third and fourth degree curves. 



(c) Proof should follow the Path of Presentment. — Of' 

 course, any valid proof will suffice for logical purposes ; but 

 when a teacher offers proof, he is trying to educate as well as 

 prove, and therefore he constantly bears in mind presentments 

 that have been made or are going to be made. For instance, 

 Euclid's 1.47 is an ideally perfect proof, because it anticipates 

 the presentment, hereafter to be shown by proportion, that 

 in a right-angled triangle each of tlie sides is a mean propor- 

 tional between its projection on the hypotenuse and the hypo- 

 tenuse itself. Other proofs of this theorem by dissection, etc., 

 are interesting and ingenious, but do not look beyond themselves. 



(d) Presentment along zeith Proof should eome as early 

 as possible in the ehain of reasoning. — See the judicious appen- 

 dix to Euclid's 1.32, added by Simson, containing the well-known 

 universal statements about all rectilineal figures. A still better 

 example is Euclid's I./ — a i)ro])osition most unaccountably re- 

 jected in some modern text-books in favour of a clumsy and 

 uninteresting proof of 1.8. The reason of this rejection, they 

 say, is " because I./ is only used to prove 1.8." But surely 

 the meaning of 1. 7 is to assert the unique pro])erty of the 

 triangle. tIc, that alone of all rectilineal figures it is unalterable 

 in shape as long as its sides remain the same. The property 

 is, moreover, of immense importance in mechanics. And it 

 becomes possible to present this ])roperty as soon .as we know 

 what a symmetrical triangle is. The property is true because 

 two symmetrical triangles cannot stand askew on the same base. 

 As soon as we know that proi^erty, 1.8 needs no further proof. 



(e) Appropriate Presentment often makes Proof Axio- 

 matie. — This we have just seen in the previous paragraph. And 

 this is the best kind of proof, enlightening the intelligence 



