Harmonic Sets and Projectivity Exercise 4.37
Besides language and music, [mathematics] is one of the primary manifestations of the free creative powers of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.
—Hermann Weyl (1885–1955)

Theorem 4.14. There exists a unique projectivity between any two harmonic sets.

Exercise 4.37. Prove Theorem 4.14. 

Proof. Assume A,B,C,D are four elements of a pencil of points that form a harmonic set H(AB,CD), and A',B',C',D' are four elements of another pencil of points that form a harmonic set H(A'B',C'D'). By the Fundamental Theorem, there exists a unique projectivity . Let D_im be the image of D under the projectivity. By the previous theorem, H(A'B',C'D_im). Thus, since the harmonic conjugate of C'  with respect to A' and B' is unique (Theorem 7), we must have D' = D_im. Therefore, there exists a unique projectivity between any two harmonic sets.//
     

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Solutions to Chapter 4

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