*45.* Writing the last equation in the form

_CB : AB = -CD : AD,_

and using the fundamental relation connecting three points on a line,

_PR + RQ = PQ,_

which holds for all positions of the three points if account be taken of
the sign of the segments, the last proportion may be written

_(CB - BA) : AB = -(CA - DA) : AD,_

or

_(AB - AC) : AB = (AC - AD) : AD;_

so that _AB_, _AC_, and _AD_ are three quantities in hamonic progression,
since the difference between the first and second is to the first as the
difference between the second and third is to the third. Also, from this
last proportion comes the familiar relation

[formula]

which is convenient for the computation of the distance _AD_ when _AB_ and
_AC_ are given numerically.