Students often associate algorithms with computers and see them as opposed to geometry and proofs. However, algorithms have always been an important part of mathematics, long before the advent of computers. Geometry and visual thinking can help in the construction and understanding of algorithms. Creating an algorithm can also be a way of proving a mathematical theorem. In this chapter, we use a few examples to illustrate these historical, geometric and ‘proving’ aspects of algorithms: the calculation by hand of the digits of a square root and the algorithm for constructing an Eulerian circuit in a multigraph in which the degree of each vertex is even.