CH. XII] MISCELLANEOUS PROBLEMS 263 



tion that we can draw a semicircle whose centre and one extremity- 

 are given, a result he had previously established. In each case 

 the demonstration is straightforward. Other solutions of this 

 proposition have been given, some of which evade the use of a 

 semicircle. 



One of his constructions is as follows. With B as centre, and 

 a radius BA, describe a semicircle of which A and G are the 

 extremities. With centres A and G, and radii AB and CA, 

 describe circles which cut in P and Q. With centres P and Q, 

 and a radius equal to AB, describe circles. These will cut in a 

 point midway between A and B. 



Here is another of his solutions, which for some purposes he 

 preferred. With B as centre, and radius BA, describe a semi- 

 circle of which A and G are the extremities. With A and G as 

 centres, and a radius equal to AB, describe circles which cut the 

 semicircle in H and K respectively. With A and G as centres, 

 and a radius equal to AG, describe circles which cut the last 

 mentioned circles (above AG) in Q and P respectively. With 

 centres P and C, and radii PA and PQ, describe circles. These 

 will cut in a point midway between A and B. 



Numerous geometrical recreations of this kind can be made 

 by any one, for all that is necessary is to select at random one 

 of Mascheroni's propositions, and see how it can be established 

 by using only circles. As instances, I select the construction on 

 a given line of a triangle similar to a given triangle (prop. 125); 

 and the construction of a regular pentagon of given dimensions 

 (prop. 137). Whatever be the solution obtained, it is always 

 interesting to turn to Mascheroni's book, and see how the question 

 is tackled there. 



