In the 14th century, Nicole Oresme first represented variability. He graphed the velocity of motion by using geometrical figures as images of different variations in motions and their magnitudes. This use of geometrical figures to depict functional relations between the time and magnitude of motion marks a significant level of abstraction.
A paper written by Leonhard Euler on the Seven Bridges of Knigsberg and published in 1741 is regarded as the first description in the history of graph theory. In Knigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.
Euler’s formula relating the number of edges, vertices, and faces of a convex polyhedron is at the origin of topology, and represents another step in abstraction.
The invention of Cartesian coordinates in the 17th century by Rene Descartes revolutionized mathematics by providing the first link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes are described with Cartesian equations. A Cartesian graph is a collection of points in the Cartesian plane. Each point in the Cartesian plane is an ordered pair of points (x, y) on the x- and y-axes. An equation with two variables is a Cartesian graph, like these two familiar examples:
- y = ax + b is a line
x^2 + y^2 = r^2 is a circle of radius r with a center at the origin.
In 1929, Frigyes Karinthy published a short story, Chain-Links. The story investigated many of the problems that would appeal to future generations of mathematicians, sociologists, and physicists. In particular, Karinthy believed that the modern world was shrinking because of the increasing connectedness of human beings. He posited that despite physical distances between individuals, the growing density of human networks made actual social distance smaller.
As a result of this hypothesis, Karinthy believed that any two individuals could be connected through at most five acquaintances. This has become known as the shortest path problem or the problem of finding a path between two vertices such that the sum of the weights of its constituent edges is minimized.