_Given four tangents and the point of contact on any one of them, to construct other tangents to a conic. Given three tangents and the points of contact of any two of them, to construct other tangents to a conic._ *91. Harmonic tangents.* We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows. It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points. It is possible, therefore, to make the following definition: _Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points._ *92. Projectivity and perspectivity.* This definition suggests the possibility of defining a projective correspondence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed. In particular, the points on a tangent are said to be _perspectively related_ to the tangents of a conic when each point lies on the tangent which corresponds to it. These notions are of importance in the higher developments of the subject. [Figure 25] FIG. 25 *93.* Brianchon’s theorem may also be applied to a degenerate conic made up of two points and the lines through them. Thus(Fig. 25), _If __a__, __b__, __c__ are three lines through a point __S__, and __a’__, __b’__, __c’__ are three lines through another point __S’__, then the lines __l = (ab’, a’b)__, __m = (bc’, b’c)__, and __n = (ca’, c’a)__ all meet in a point._ *94. Law of duality.* The observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding. He will have noted that points have replaced lines and lines have replaced points; that points on a curve have been replaced by tangents to a curve; that pencils have been replaced by point-rows, and that a conic considered as made up of a succession of points has been replaced by a conic considered as generated by a moving tangent line. The theory upon which this wonderful _law of duality_ is based will be developed in the next chapter. PROBLEMS 1. Given four lines in the plane, to construct another which shall meet them in four harmonic points. 2. Where are all such lines found? 3. Given any five lines in the plane, construct on each the point of contact with the conic tangent to them all. 4. Given four lines and the point of contact on one, to construct the conic. ("To construct the conic" means here to draw as many other tangents as may be desired.) 5. Given three lines and the point of contact on two of them, to construct the conic. 6. Given four lines and the line at infinity, to construct the conic. 7. Given three lines and the line at infinity, together with the point of contact at infinity, to construct the conic. 8. Given three lines, two of which are asymptotes, to construct the conic.