Mr. T. S. Davies on Geotnetry and Geometers. 503 



fi{n 1) 



two, say «, then there could be -^-^ — perpendiculars drawn 



from C to AB. The line FG must then fall taholly 'within 

 the circle; whereas when we come to the third book we are 

 required to prove this very property ; and more strano^ely still, 

 to draw a straight line isoithin the circle in the antecedent pro- 

 position. The difficulty is inherent in the method pursued. 

 // compels the eye to supply the place of reason. We merely see 

 that it is : — not know why it must be so. It is not by such 

 phrases as "who is so dull as not to see?" or "it is easy to 

 see," that this blemish can be removed. 



I believe that no one who has been much concerned in 

 mathematical education will complain of students " only be- 

 ginning the Elements " being very short-sighted upon such 

 points as these. They find it " easy to see" that many things 

 are so and so which the tutor would prefer their proving. In 

 these cases, it is not the fact that is in (juestion, but its place 

 in the logical system. Geometry by inspection is always more 

 acceptable to the majority of young students than geometry 

 by demonstration. Mr. Byrne must have known this, when 

 he carried the system to perfection in his Euclid in Colours. 

 Did geometry, however, stop at visible properties^ it would be 

 jejune enough. I have often found, too, that the notion of 

 the ultimate objects of geometrical research being principally 

 graphic, or in some other way merely practical, has, whilst it 

 has fostered the propensity to proof by inspection, generally 

 led to much confusion in the student's mind. Let it always 

 be presented to his mind as a rational science — never as a 

 practical art, except, indeed, incidentally. 



" 4. In the same place he blames the bidding ' draw a straight 

 line within a circle, without specifying that it must terminate in the 

 circumference ' not knowing that this way of speaking is constantly 

 used by Euclid and other Geometers ; as not only in this P*. of the 

 3^ but in tlie 4, 14, 15, 28 of the same book, and though the 2 Prop, 

 of the 3*^ is to prove that a straight line joining any two points of 

 the circumference is within the circle, yet surely there is no need of 

 demonstrating that a straight line may be drawn somewhere within 

 the circle ; the 2"* Prop, being to shew that though the points be 

 taken never so near to one another, that the straight line joining 

 them will neither fall upon nor without the circumference," 



Simpson had only noticed this as an instance of vague ex- 

 pression. Is it not such? Is it not more — even much more? 



" 5. In the same note he answers what is said in page 415. of the 

 4*° English Edition, that it ought to be demonstrated not assumed 

 that a straight line cannot meet a straight line in more than one 

 point, and says it cannot be demonstrated, and that the Editor refers 



