52 GEOMETRICAL RECREATIONS [CH. Ill 



of the abscissa by the formula r = a + ex. Hence drjdx = e. 

 From this it follows that r cannot have a maximum or minimum 

 value. But the only closed curve in which the radius vector 

 has not a maximum or minimum value is a circle. Hence, every 

 ellipse is a circle, a result which obviously is untrue. 



Geometrical Paradoxes. To the above examples I may 

 add the following questions, which, though not exactly falla- 

 cious, lead to results which at a hasty glance appear impossible. 



First Paradox. The first is a problem, sent to me by 

 Mr W. Renton, to rotate a plane lamina (say, for instance, a 

 sheet of paper) through four right angles so that the effect is 

 equivalent to turning it through only one right angle. 



Second Paradox. As in arithmetic, so in geometry, the 

 theory of probability lends itself to numerous paradoxes. 

 Here is a very simple illustration. A stick is broken at 

 random into three pieces. It is possible to put them together 

 into the shape of a triangle provided the length of the 

 longest piece is less than the sum of the other two pieces 

 (cf. Euc. I. 20), that is, provided the length of the longest 

 piece is less than half the length of the stick. But the 

 probability that a fragment of a stick shall be half the 

 original length of the stick is 1/2. Hence the probability that 

 a triangle can be constructed out of the three pieces into 

 which the stick is broken would appear to be 1/2. This is not 

 true, for actually the probability is 1/4. 



Third Paradox. The following example illustrates how 

 easily the eye may be deceived in demonstrations obtained by 

 actually dissecting the figures and re-arranging the parts. In 

 fact proofs by superposition should be regarded with consider- 

 able distrust unless they are supplemented by mathematical 

 reasoning. The well-known proofs of the propositions Euclid 

 I. 32 and Euclid I. 47 can be so supplemented and are valid. 

 On the other hand, as an illustration of how deceptive a non- 

 mathematical proof may be, I here mention the familiar paradox 

 that a square of paper, subdivided like a chessboard into 

 64 small squares, can be cut into four pieces which being put 



