112. We have next to consider the quadrilateral construction[124]. This has a double purpose: first, to define the important special case known as a harmonic range; and secondly, to afford an unambiguous and exhaustive method of assigning different numbers to different points. This last method has, again, a double purpose: first, the purpose of giving a convenient symbolism for describing and distinguishing different points, and of thus affording a means for the introduction of analysis; and secondly, of so assigning these numbers that, if they had the ordinary metrical significance, as distances from some point on the numbered straight line, they would yield –1 as the anharmonic ratio of a harmonic range, and that, if four points have the same anharmonic ratio as four others, so have the corresponding numbers. This last purpose is due to purely technical motives: it avoids the confusion with our preconceptions which would result from any other value for a harmonic range; it allows us, when metrical interpretations of projective results are desired, to make these interpretations without tedious numerical transformations, and it enables us to perform projective transformations by algebraical methods. At the same time, from the strictly projective point of view, as observed above, the numbers introduced have a purely conventional meaning; and until we pass to metrical Geometry, no reason can be shown for assigning the value –1 to a harmonic range. With this preliminary, let us see in what the quadrilateral construction consists.
113. A harmonic range, in elementary Geometry, is one whose anharmonic ratio is –1, or one in which the three segments formed by the four points are in harmonic progression, or again, one in which the ratio of the two internal segments is equal to the ratio of the two external segments. If a, b, c, d be the four points, it is easily seen that these definitions are equivalent to one another: they give respectively:
ab bc / ad dc = – 1 , 1 ab – 1 ac = 1 ac – 1 ad , and ab bc = ad cd .
But as they are all quantitative, they cannot be used for our present purpose. Nor are any definitions which involve bisection of lines or angles available. We must have a definition which proceeds entirely by the help of straight lines and points, without measurement of distances or angles. Now from the above definitions of a harmonic range, we see that a, b, c, d have the same anharmonic ratio as c, b, a, d. This gives us the property we require for our definition. For it shows that, in a harmonic range, we can find a projective transformation which will interchange a and c. This is a necessary and sufficient condition for a harmonic range, and the quadrilateral construction is the general method for giving effect to it.
Given any three points A, B, D in one straight line, the quadrilateral construction finds the point C harmonic to A with respect to B, D by the following method: Take any point O outside the straight line ABD, and join it to B and D. Through A draw any straight line cutting OD, OB in P and Q. Join DQ, BP, and let them intersect in R. Join OR, and let OR meet ABD in C. Then C is the point required.
To prove this, let DRQ meet OA in T, and draw AR, meeting OD in S. Then a projective transformation of A, B, C, D from R on to OD gives the points S, P, O, D, which, projected from A on to DQ, give R, Q, T, D. But these again, projected from O on to ABD, give C, B, A, D. Hence A, B, C, D can be projectively transformed into C, B, A, D, and therefore form a harmonic range. From this point, the proof that the construction is unique and general follows simply[125].