CH. Ill] GEOMETRICAL RECREATIONS 51 



Therefore AD and BO, which are perpendicular to this direc- 

 tion, must be parallel. This result is not universally true, 

 and the above demonstration contains a flaw. 



Seventh Fallacy. The following argument is taken from a 

 text-book on electricity, published in 1889 by two distinguished 

 mathematicians, in which it was presented as valid. A given 

 vector OP of length I can be resolved in an infinite number 

 of ways into two vectors OM, MP, of lengths V, I", and we 

 can make I'll" have any value we please from nothing to 

 infinity. Suppose that the system is referred to rectangular 

 axes Ox, Oy; and that OP, OM, MP make respectively angles 

 6, 6', 6" with Ox. Hence, by projection on Oy and on Ox, 

 we have 



Jsin0 = Z'sin0' + rsin0", 



J cos = J' cos 0' + J" cos 0". 

 a — n s * n ^ "*" s * n ^" 



• ■ tan V — jr. ■ jrr, . 



ncosff +cos0 



where n = I'/l". This result is true whatever be the value of n. 

 But n may have any value {ex. gr. n = oo , or n = 0), hence 

 tan = tan 6' = tan 6", which obviously is impossible. 



Eighth Fallacy*. Here is a fallacious investigation of the 

 value of 7r: it is founded on well-known quadratures. The 

 area of the semi-ellipse bounded by the minor axis is (in the 

 usual notation) equal to ^-rrab. If the centre is moved off to 

 an indefinitely great distance along the major axis, the ellipse 

 degenerates into a parabola, and therefore in this particular 

 limiting position the area is equal to two-thirds of the circum- 

 scribing rectangle. But the first result is true whatever be 

 the dimensions of the curve. 



,\ %Trab = %a x 2b, 



.'. 7T= 8/3, 



a result which obviously is untrue. 



Ninth Fallacy. Every ellipse is a circle. The focal distance 

 of a point on an ellipse is given (in the usual notation) in terms 



* This was communicated to me by Mr R. Chartres. 



4—2 



