Manchester Memoirs \ Vol. lix. (19 15), No. 13. 7 



A draw a straight line intersecting CB at D and BG at 

 N, so that the intercept DH shall be equal to twice AB. 

 This problem was solved by Pappus in two ways : (1) by 

 using a conchoid, (2) by using a rectangular hyperbola. 

 The conchoid used was the locus of a point P {Fig. 5) 

 obtained by drawing any straight line through A inter- 

 secting BG at Q and making QP equal to a constant 

 length equal to twice AB, The intersection of this con- 

 choid with CB is the required point D, the angle ADB, 

 or «, being one-third of the angle ABC. This becomes 

 obvious if E s the middle point of DH, be joined to B. 



The other method used by Pappus was to draw a 

 rectangular hyperbola passing through B {Fig. 6) with 



G K 



Fig. 6. Method of Pappus (Use of Hyperbola). 



M 



asymptotes AM and AN parallel to BC and BG respec- 

 tively. This will be the locus of a point X where AQP 

 is any straight line passing through A, and OX and PX 

 are parallel to' BC and BG. A circle with B as centre 

 and radius twice AB intersects the hyperbola at F. In- 

 completing the rectangle BHFD on BF as diagonal, the 

 required points // and 1) are found, 111) being equal to 



