Karl Georg Christian Von Staudt - Algebra of Throws

In 1857, in the second Beiträge, von Staudt contributed a route to number through geometry called the Algebra of throws (German: Wurftheorie). It is based projective range and the relation of projective harmonic conjugates. Through operations of addition of points and multiplication of points, one obtains an "algebra of points", as in chapter 6 of Veblen & Young's textbook on projective geometry. The usual presentation relies on cross ratio (CA,BD) of four collinear points. For instance, Coolidge (1940) wrote:

How do we add two distances together? We give them the same starting point, find the point midway between their terminal points, that is to say, the harmonic conjugate of infinity with regard to their terminal points, and then find the harmonic conjugate of the initial point with regard to this mid-point and infinity. Generalizing this, if we wish to add throws (CA,BD) and (CA,BD' ), we find M the harmonic conjugate of C with regard to D and D', and then S the harmonic conjugate of A with regard to C and M :

In the same way we may find a definition of the product of two throws. As the product of two numbers bears the same ratio to one of them as the other bears to unity, the ratio of two numbers is the cross ratio which they as a pair bear to infinity and zero, so Von Staudt, in the previous notation, defines the product of two throws by

These definitions involve a long series of steps to show that the algebra so defined obeys the usual commutative, associative, and distributive laws, and that there are no divisors of zero.

A summary statement is given by Veblen & Young as Theorem 10: "The set of points on a line, with removed, forms a field with respect to the operations previously defined". As Freudenthal notes

...up to Hilbert, there is no other example for such a direct derivation of the algebraic laws from geometric axioms as found in von Staudt's Beiträge.

Another affirmation of von Staudt's work with the harmonic conjugates comes in the form of a theorem:

The only one-to-one correspondence between the real points on a line which preserves the harmonic relation between four points is a non-singular projectivity.