MATHEMATICAL SECTION. 
99 
(5) 
, /l/o , 
T-hr—^ + 
l/l — ^ 
) 
( 6 ) 
— 4x^ + ^ — 2 /) 
( 3 ?/“ 4 /;^ = 
2 ) 
Ihid. 
Ibid. 
Dr. William Hillhouse, of New Haven, has not only shown that 
the last five curves may be used for trisecting an angle, but has 
devised instruments for describing them. This latter step, as I 
view the problem of trisection, is a necessary part of any real solu¬ 
tion by means of curves. The simple determination of the equations 
that will satisfy the analytical conditions of trisection, wdthout the 
instrumental means for describing the curves, does not constitute a 
solution of the problem. It is essentially something to be done, not 
something merely to be proved, and its fame arises from the im¬ 
possibility of making the trisection of an angle by means of a ruler 
and pair of compasses. Any proposed solution, therefore, must in¬ 
dicate a geometrical instrument for accomplishing it, and the nature 
of the motion applied therein determines the character of the solu¬ 
tion. 
A class of solutions, regarded the most elegant of all, is that in 
which link machines trisect the angle directly, without the aid of 
interposed curves. Of this class I have found three instruments, 
invented respectively by the Marquis de I’Hopital (1661-1704), 
A. B. Kempe, and Professor Sylvester. That of THopital is a com¬ 
bination of link and sliding motion ; those of Kempe and Sylvester 
nre pure linkages. With respect to the two latter, I wish to call 
attention to the elementary character of the solution. Euclid’s 
postulates require us to be able to draw a straight line and a circle, 
and it is frequently assumed that they imply the use of a straight¬ 
edge and a pair of compasses. But, manifestly, any other instru¬ 
ments that can describe the circle and the straight line satisfy 
equally the requirements of the postulates. Moreover, the use of a 
straight edge assumes as accomplished the very thing proposed to 
be done; whence the straight-edge is not an original instrument 
for describing a straight line. Such an instrument is given by the 
linkage of Peaucellier. More originally, therefore, than by the 
straight-edge, the postulates of Euclid may be assumed to imply 
the use of pure link motion. Wherefore, if pure link motion be 
considered as postulated by Euclidian geometry, the trisection of 
an angle becomes one of the simplest of geometrical problems. 
