262 MISCELLANEOUS PROBLEMS [CH. XII 



quadrant and the top half of the third quadrant. In the third 

 card, we get two windows by cutting out the right half of the 

 second quadrant and the right half of the third quadrant. In 

 the fourth card, we divide the second and third quadrants into 

 four equal horizontal strips and from each of these quadrants 

 cut out the first and third strips. In the fifth card, we divide the 

 second and third quadrants into four equal vertical strips, and 

 from each of these quadrants cut out the second and fourth of 

 these strips. In the sixth card we divide the second and third 

 quadrants into eight horizontal strips and from each of these 

 quadrants cut out the first, third, fifth, and seventh strips. 



It will be noticed that no windows are cut in the first or 

 fourth quarter of any card; hence they are free for insertion of 

 the 64 numbers written on the face of each card. The construc- 

 tion here given is due to my friend R. A. L. Cole. 



Possibly the puzzle is better presented by omitting all 

 numbers exceeding 100, for the introduction of 128 at once 

 suggests the method of construction. With that restriction I 

 think the use of only seven cards is better and more elegant than 

 the form in which I have seen it on sale. 



Compass Problems. It is well known that Euclid in his 

 Elements confined his constructions to those which could be 

 made with ungraduated rulers and compasses. The use of a ruler 

 is however unnecessary in many cases, and Mascheroni* estab- 

 lished a number of propositions by constructions made with 

 compasses alone. He presented his propositions as a connected 

 series, and of course the logical sequence is very different from 

 that with which we are familiar. It is remarkable how much he 

 effected without being able to join points by straight lines or 

 make use of properties of parallels. 



As an instance which will illustrate the subject, I select the 

 problem to find a point midway between two given points A and 

 B. Of this fundamental proposition Mascheroni gave five solu- 

 tions (prop. 66). Here are two of them : they rest on the assump- 



* His work was first published at Bergamo, 1782. French translations were 

 issued at Paris in 1798 and 1828, and from these my knowledge of the book is 

 derived. 



