280 REV. T. p. KIEKMAN ON LFNEAR CONSTRUCTIONS. 



What mathematicians have been so long looking for, is 

 a property in solid geometry like the celebrated theorem 

 of Pascal. By this, if any five points be given in aplane, and 

 any line through one of them, we can determine by the 

 joining of given points, that is, with the ruler only, where the 

 given line cuts a second time the conic which passes through 

 the five points. Pascal's theorem requires only a ruler; but 

 it is important to observe, that there is no limit to the 

 length of that ruler. If the given points are 12 3 4 5, 

 and A 1 be the given line, Pascal's theorem teaches us, 

 considering the hexagon 12 3 4 5 6 — 6 being the sought 

 point in A 1 — to produce the line 34 to meet A 1, and 12 

 to meet 45 ; then, through the points (A 1, 34) and (12, 

 45) thus found, to draw a line cutting 23 in a point from 

 which a Hne drawn through 5 will cut A 1 in the point 6 

 required. If now it happens, either that A 1 is parallel to 

 34, or that 12 is parallel to 45, the line cutting 23 cannot 

 be drawn with a ruler of finite length, since one of its two 

 points (A 1, 34) and (12, 45) passes off to an infinite dis- 

 tance. Thus, Pascal's theorem itself can be shown to fail 

 unless an infinite ruler be conceded, that is, unless it is 

 granted that a line is given by its direction and one of its 

 points, or that we have the power of drawing parallels. 

 Nor is it any answer to this claim to say, that we can 

 choose another hexagon. For how far are we to pursue 

 the lines Al and 34, in order to convince ourselves that 

 they never meet ? 



Let but a ruler of imlimited length be granted me, and 

 I will show how to effect, by its aid only, and from purely 

 geometrical data, the solution of a more general question 

 than the Brussels prize question, namely, this problem of 

 linear constructions : — 



Any locus of the N'^ order (or class) being given geometri' 

 tally y by the requisite number of points (or tangents) of which 

 n-\ are in a line (or pass through a point), to find the N"" point 



