218 Transactions. 



ago both sailors used a similar implement to fasten planks together with 

 cords. European and Maori sailors alike used it mth its holes for reeving 

 purposes. 



As the various tribes of Maoris used the same instrument, it is clear that 

 the names and the implement itself were known in the ancient Hawaiki. 

 It will also be seen that its names aliao and purupuru refer to the two main 

 purposes to which the tool was put. 



Mr. Percy Smith recognised the implement, having known similar ones 

 years ago. He thought it was called a Jcaneka or taneka, words which have 

 not been preserved in Maori dictionaries, the idea being suggestive of the 

 word " piercer," which would suggest one of its uses. Aneanc or aneha 

 means " sharp-pointed." A tao or hao is a sharp-pointed spear. Thus in 

 the various Maori dialects this instrument was called by many names. 



This paper, short as it is, embodies a considerable amount of research, 

 I having ransacked much literature and consulted many experts in Maori 

 lore, to many of whom the implement was quite unknown. I think this 

 embodies all that will ever be discovered about the ahao. 



Art. XXX. — On the Trisection of an Angle. 



By H. W. Segar, M.A., Professor of Mathematics, Auckland Univer- 

 sity College. 



[Read before the Auckland Institute, 11th August, 1908.1 



Three problems of an apparently simple character are famous as having 

 engaged the attention of the ancients, who sought in vain for solutions 

 by means of the straight-edge and compass. These were (1) the 

 trisection of an arbitrary angle ; (2) the quadrature of the circle ; (3) the 

 duplication of the cube. 



These have generally ceased now to be objects of attack by mathe- 

 maticians, because it is now known that it is impossible to solve any one 

 of these problems by means of the straight-edge and compass. The 

 proofs of this impossibility are available to the English reader in the 

 translation of Klein's "Famous Problems in Elementary Geometry," 

 published by Ginn and Company. Occasionally, however, one or other 

 of these problems takes possession of some person unaware of these 

 investigations, and with only a slight knowledge of mathematics. I was 

 recently approached by Mr. Viggo Hansen, of this city, who submitted to 

 me what he considered was a solution of the first problem — the ti'isection 

 of an angle. On testing his drawing I could discover no inequality in the 

 three parts into which his construction had divided the original angle. 

 Thinking that the construction happened to suit the particular angle 

 chosen, I carried it out for three other angles very different in magnitude 

 from one another and from Mr. Hansen's, with exactly the same result. 

 It was evident then that Mr. Hansen had obtained an approximate 

 solution of the problem of exceptional interest, especially as the con- 

 struction is very simple, and depends on the bisection of angles. The 

 construction was as follows : — 



