=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. [Illustration] =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. [Illustration]