and _L_. Join _N_ and _L_ to _C_ and get the points _K_ and _M_ on the two
lines through _A_. Then _KM_ is parallel to _AC_. _The bisection of a
given segment and the drawing of a line parallel to the segment are
equivalent data when linear construction is used._




*42.* It is not difficult to give a linear construction for the problem
to divide a given segment into _n_ equal parts, given only a parallel to
the segment. This is simple enough when _n_ is a power of _2_. For any
other number, such as _29_, divide any segment on the line parallel to
_AC_ into _32_ equal parts, by a repetition of the process just described.
Take _29_ of these, and join the first to _A_ and the last to _C_. Let
these joining lines meet in _S_. Join _S_ to all the other points. Other
problems, of a similar sort, are given at the end of the chapter.




*43. Numerical relations.* Since three points, given in order, are
sufficient to determine a fourth, as explained above, it ought to be
possible to reproduce the process numerically in view of the one-to-one
correspondence which exists between points on a line and numbers; a
correspondence which, to be sure, we have not established here, but which
is discussed in any treatise on the theory of point sets. We proceed to
discover what relation between four numbers corresponds to the harmonic
relation between four points.

[Figure 8]