Most people who have studied some physics or chemistry know that it is important to keep the units of our numbers straight when we do calculations. Failure to attend to units usually leads to wrong answers.
What is not well known is that the analysis of units can often help scientists to derive formulas, even when the underlying physics is not well understood. How it works seems a bit mysterious, and the technique was not understood or appreciated until about 1870, when the great physicist James Clerk Maxwell laid out the principles of the technique, which is formally known as Dimensional Analysis.

The history of mathematics has many instances where someone has posed a problem for the mathematical world at large to solve, and the problem was not resolved for decades, or even centuries. Often, new mathematics has been discovered in the process of working out a solution.
This post is the story of one such case, the so-called Basel Problem, first posed as a challenge to European mathematicians in 1644. It withstood all attempts to solve it until, in 1734, young Leonard Euler found the answer. As the reader will see, Euler’s solution is a work of astonishing ingenuity, even though the level of the mathematics does not go beyond Algebra I.