> In the James Smart text 'Modern Geometries' on page > 214 fig 5.4 A Harmonic set of points is defined as > follows: > "In Euclidean Geometry, a harmonic set of points (or > harmonic range) is defined to be a set of four > collinear points such that the segment determined by > two of them is divided internally and externally in > the same ratio." > > To me this is not plain English. What does it mean > to say: "the segment determined by two of them?" And > mainly I don't understand this concept of dividing a > segment externally.
suppose the points in increasing order are a,b,c,d. Then b divides [a,c] internally in the ratio  (b-a)/(c-b). And d divides [a,c] externally in the ratio  (d-a)/(d-b). For certain choices of a,b,c,d. the ratios  and  might be the same. In that case one has a harmonic set of points.