Dimensional analysis

Basically, dimensional analysis is a method for reducing the number and complexity of experimental variables which affect a given physical phenomenon, by using a sortof compacting technique. If a phenomenon depends upon n dimensional variables, dimensional analysis will reduce the problem to only k dimensionless variables, where the reduction n-k=1, 2, 3, or 4, depending upon the problem complexity. Generally
n-k equals the number of different dimensions (sometimes called basic or primary or fundamental dimensions) which govern the problem.

Historically, the first person to write extensively about units and dimensional reasoning in physical relations was Euler in 1765. Euler’s ideas were far ahead of his time, as were those of Joseph Fourier, whose 1822 book "Analytical Theory of Heat" outlined what is now called the principle of dimensional homogeneity and even developed some similarity rules for heat flow. There were no further significant advances until Lord Rayleigh’s book in 1877, "Theory of Sound", which proposed a “method of dimensions” and gave several examples of dimensional analysis. The final breakthrough which established the method as we know it today is generally credited to E. Buckingham in 1914, whose paper outlined what is now called the Buckingham pi theorem for describing dimensionless parameters. However, it is now known that a Frenchman, A. Vaschy, in 1892 and a Russian, D. Riabouchinsky, in 1911 had independently published papers reporting results equivalent to the pi theorem. Following Buckingham’s paper, P. W. Bridgman published a classic book in 1922, outlining the general theory of dimensional analysis. The subject continues to be controversial because there is so much art and subtlety in using dimensional analysis.