The Trisection of an Angle. 



BY A. h. CANDY. 



The ancient geometricians were chiefly concerned with three 

 problems; namely, the duplication of a cube, the trisection of an 

 angle, and the squaring of a circle. Hippocrates of Chios (born 

 470 B. C.) reduced the first of these to that of finding two geometric 

 means, x and y, between one straight line a and another twice as 

 long 2a; for if a : x=ix : y=y : 2a, then x^:=-2a.^; but he failed to 

 find these means. It was subsequently proved (probably about 1600 

 A. D. ), by \'ieta, a French mathematician, that the first two of 

 these problems, considered analytically, require the solution of a 

 cubic equation; and since a construction by means of circles, whose 

 equations are of the form x^-f yS-f ax-f by + c=o, and straight lines, 

 whose equations are of the form a'x-j-b'y-f c'=o, is only equivalent 

 to the solution of a quadratic or a biquadratic equation, these 

 problems are not soluble if we are restricted to lines and circles. In 

 the second century A. D., Nicomedes invented the Conchoid and 

 Diodes invented the Cissoid, both curves being used to give a solu- 

 tion of the duplication problem, and probably the former was 

 also used to trisect an angle. If this property of the curve was known 

 to Nicomedes it was doubtless a part of the cause that led to its 

 invention. Be this as it may, in 1677 Viviani, a pupil of Galileo, 

 showed how this problem could be solved by the aid of t'.iis curve; 

 he also showed how an angle could be trisected by the aid of the 

 equilateral hyperbola. Such is the brief history of these famous 

 problems. 



Modern mathematicians have not considered them of sufificient 

 importance to merit special attention. Indeed it is not now claimed 

 that the trisection of an angle is of any practical value or of much 

 interest save that which clusters around its history; but it is believed 

 that, probably, at least three -of the solutions here given are new. 

 The solutions by means of the hyperbola are possibly old; if so, their 

 publication in this series is justified by their intimate relation 

 to the other methods; if not, their promulgation calls for no apology. 



All the methods of solution herein presented are based upon a 

 simple theorem in Elementary Geometry and developed from it 



{2b) KAN. UNIV. QUAR., VOL. II, NO. 1, JULY, 1893. 



