48 Traces of Primitive Man. 



cases the division of the angle precedes the division of the arc ; 

 in other cases the arc is divided first, and then the division of 

 the angle is derived from the division of the arc. For example, 

 in one of those instruments of which I shall presently describe 

 the operation, that for the trisection of an angle, it is the arc 

 which is mechanically trisected by pointing off its third part on 

 the circle ; while another instrument which I shall describe, 

 being for the construction of a particular triangle, must have its 

 result applied to a circle, if the arc answering to its angles is to 

 be divided. 



Much attention was paid by the geometers to the trisection 

 of angles. It is almost if not quite self-evident that the bisec- 

 tion of angles must have been very early practised, the operation 

 being most simple, and the proof equally simple. It is probable 

 that many generations of mathematicians were ignorant that 

 the trisection of any angle, excepting always angle 90° and 

 its half, quarter, &c., is impossible in plane geometry, and it is 

 probable that after the impossibility of that operation was con- 

 sidered to be established, the analytical reason of this continued 

 long to be unknown. Plane geometry admits only of the 

 solution of problems, the algebraic equations whereof, under any 

 algebraic alphabet, are expressible in equations of the second 

 degree. An equation of the third degree is the symbol of and 

 represents actually or by inference a solid, or something derived 

 from a solid, and plane geometry does not take cognisance of 

 more than two dimensions. The equation of a straight line in 

 relation to the circle, and the converse equation of a circle in 

 relation to a straight line, are of the second degree ; the 

 equations of all other except a limited number of lines — we may 

 regard the circle as a line if we consider its circumference only — 

 are of the third and higher degrees. It has come from this to be 

 a maxim that in plane geometry no instruments are admissable 

 for use except the rule and the compasses, which, however, is 

 not a principle, but rather a result from the analytical reason I 

 have spoken of. This leads me to observe, by way of parenthesis 

 — for this is not directly associated with my proper subject — 



