1901-2.] Prof. Chrystal on Theory of Miller’s Trisector. 
9 
Note on the Mathematical Theory of Miller’s Trisector, 
and its Relation to other Solutions of the Problem of 
Trisection. By Professor Chrystal. 
(Read November 4, 1901. Issued separately February 12, 1902.) 
The problem of the trisection of an angle, analytically considered, 
depends on the solution of a cubic equation, say 4x 8 - 3x — cos a = 0, 
which is in general irreducible, and therefore not soluble by means 
of quadratic radicals. It follows that the trisection of an angle 
cannot be effected by means of the ruler and compass alone. This, 
in fact if not in theory, appears to have been known to the early 
Greek geometers, and they proposed the use of various curves, the 
continuous mechanical construction of which must, of course, be 
postulated, for the solution of the problem. The Quadratrix and 
the Spiral of Archimedes, both transcendental curves, may be 
mentioned. It was early discovered that the conic sections could 
be used for the purpose, and in modern times various solutions have 
been suggested involving their use.* 
The earliest example of the use of an algebraic curve of higher 
degree than the second is the trisection by means of the conchoid 
of Nicomedes, a circular quartic, represented by the equation 
(x 2 + y 2 )(x - a) 2 = b 2 x 2 . This solution is of special interest, because 
the conchoid admits of a very simple mechanical construction by 
means of the well-known Trammel of Nicomedes. If, as is commonly 
believed, this apparatus was invented by Nicomedes himself (< ca . 
150 b.c.), this would be the earliest example of the mechanical 
construction of a curve of higher degree than the second. 
Among the trisectrix curves of higher degree that have been 
used by modern mathematicians may be mentioned the Cubic 
* Those interested in the subject may consult Cantor’s Geschichte der Mathe- 
matik (Leipzig, 1894) ; Allman’s Greek Geometry (Dublin, 1889) ; Gow’s Short 
History of Greek Mathematics (Cambridge, 1884) ; Newton’s Arithmetica Uni- 
versalis, 2nd ed. (London, 1722), Appendix de Equationum Constructione 
Lineari ; Maclaurin’s Algebra (London, 1756), chap. iii. ; Klein’s Vorlesungen 
iiber Ausgewahlte Fragen der Element argeomctrie (Leipzig, 1895). 
