This chapter presents a brief history of the concept of dimension and of Dimensional Analysis.

Keywords

Dimensional Analysis; unit homogeneity

2.1 Pre Joseph Fourier

The concept of dimension is as old as Greek mathematics, but the use of dimension as an analytical tool is relatively modern. Greek mathematics, i.e., geometry, is based on length and dimensionless angle. The Greeks did not consider the implications of dimension since all their mathematical manipulations involved only lengths and angles.

When our earliest ancestors learned to count is unknown, but it surely began shortly after they realized they had fingers and toes. With those fingers and toes, the rules of pure number manipulation came to light. Thus, by the time history began, our ancestors knew how to manage pure numbers; they knew the rules of arithmetic.

With the development of algebra, higher mathematics freed itself from geometry. In algebra, numbers can represent physical quantities, which have dimensions. Attaching dimensional information to numbers negates the rules of arithmetic, unless we observe certain restrictions.1

The first to discuss the concept of dimension was Johannes de Mures (c.1290–c.1355), a French philosopher, astronomer, mathematician, and music theorist. He wrote about products and quotients possessing different dimensions. However, his work on products and quotients made no lasting impression on the development of science.

Descartes (1596–1650) may have been the first natural philosopher and mathematician to realize that derived dimensions exist, such as Force.2 According to Descartes, “[t]he force to which I refer always has two dimensions, and it is not the force that resists (a weight) which has one dimension”.3 However, Descartes was not hindered by mathematical operations that produced dimensionally impossible results. For Descartes, dimensional correctness did not determine the correctness of a given result.

Sir Issac Newton (1642–1727) recognized the concept of derived dimensions: “I call any quantity a genitum which is not made by addition or subtraction of divers parts, but is generated or produced in arithmetic by multiplication, division, or extraction of the root of any term whatsoever ….”4 Gottfried Leibniz (1646–1716) also recognized the concept of derived dimensions, no doubt to Sir Issac’s chagrin: “… action … is as the product of the mass multiplied by space and velocity, or as the time multiplied by vis viva.”4

The eighteenth century witnessed great advances in analysis of physical phenomena; however, little thought was given to dimensions. Leonhard Euler (1707–1783) was the only natural philosopher and mathematician to make comment on dimensions during that momentous century. In fact, Euler demonstrated a preoccupation about the meaning of physical relationships. In 1736, Euler published Mechanica in which he showed that the dimension of n in the equation

depended on the dimensions of A (mass) and p (force). This observation by Euler indicates that he understood the need for unit homogeneity; that is, the units left of an equal sign must be the same as those units to the right of the same equal sign. Euler further discussed dimensions in his Theoria motus corporum solidorum seu rigidorum published in 1755. In this book, Euler devoted a chapter to questions of units and homogeneity. Unfortunately, his writings about dimension made little impression upon the community of mathematicians and natural philosophers of the time.

2.2 Post Joseph Fourier

Little, if any, discussion of dimensions occurred after Euler’s Theoria until 1822 when Joseph Fourier published the third edition of his Analytical Theory of Heat. Fourier makes no mention of dimension in either the first edition of his book, published in 1807, or the second edition, published in 1811. However, in the 1822 edition, Fourier specifically states that any system of units can be used to study a physical process, so long as the chosen system of units is consistent. He also states that mathematical equations used to describe physical processes must demonstrate homogeneity: the units on either side of an equal sign must be the same. Fourier used the concept of homogeneity to check his mathematical manipulations. He clearly states in the 1822 edition that a natural philosopher should use unit homogeneity as a check on his mathematical analysis of a physical process.4 If the units are not the same on either side of the final equal sign, then the natural philosopher has incorrectly manipulated a mathematical equation occurring earlier in the study—thus, the “birth” of Dimensional Analysis.

While Fourier may have published the basics of Dimensional Analysis and presented the need for unit homogeneity when investigating a physical process, few, if any, pursued his insights. In fact, confusion over dimensions and units persisted through the two middle quarters of the nineteenth century.5 It was the development of electrical technology and telegraphy, in general, and the trans-Atlantic telegraph cable, in particular, that forced a discussion and review of dimensions and units in the 1860s.6

In 1861, the British Association for the Advancement of Science (BAAS) formed a committee to review the various systems of units for electricity and magnetism in use at the time. The committee was also charged with codifying a standard system of units, especially for electrical measurements. This initiative by BAAS was the first effort by engineers and scientists to develop an understanding of units and to establish a standardized system of units.

William Thomson (later Lord Kelvin) and James Clerk Maxwell served on the committee and greatly influenced its program.7 Maxwell provided the greatest insight into the effort, but he also created the most confusion, confusion that continues to this day. Maxwell realized that physical concepts are quantified by dimensions, e.g., by Length, Mass, and Time. Various sets of dimensions could be grouped and called fundamental dimensions, from which other dimensions, such as Force, Energy, and Power, could be derived. Maxwell suggested identifying fundamental dimensions by brackets; thus, the fundamental dimensions Length, Mass, and Time would be identified as [LMT]. He did not clarify what he meant by his bracket notation; hence, confusion developed with regard to them and still exists about them today. In reality, we should consider them as identifying the procedures to be used when describing a physical concept. Once we have identified the fundamental dimensions for describing a physical concept, then we can develop a set of standards to quantify the chosen dimensions: those standards form a system of units. Many engineers and scientists expended considerable effort during the last quarter of the nineteenth century developing the standards for various systems of units.8,9

In 1877, Lord Rayleigh published his Theory of Sound. Its index contains an entry entitled “Method of Dimensions.”4 Lord Rayleigh made good use of Dimensional Analysis during his long and fruitful scientific career; however, he never presented a derivation of his method. He simply stated the method could be used as a research tool when investigating physical processes. Lord Rayleigh equated the powers or indices of the dimensions that describe a physical process. His method works well for simple mechanical processes where the number of unknown exponential indices equals the number of equations.10 For processes involving heat or mass transfer, there will be more unknowns than equations; therefore, the practitioner of Lord Rayleigh’s method must assign a value to each of these unknowns, then prove that the assumed unknowns yield an independent set of results. In other words, this method becomes cumbersome and time consuming when applied to complex physical and chemical processes.

While engineers and scientists in Great Britain used Dimensional Analysis without mathematical validation of it, their colleagues on the Continent were investigating the concept of dimension at a more philosophical level. In 1892, A. Vaschy, a French electrical engineer, published a version of what became known as Buckingham’s Pi theorem.11 In the first chapter of his Theorie de l’Electricite, published in 1896, Vaschy discusses dimensions, systems of units, and measurements.4 He presents Buckingham’s Pi theorem in modern notation in Chapter One. Unfortunately, the scientific community lost interest in Dimensional Analysis after Vaschy’s publication because no reference to Dimensional Analysis occurs until 1911.

In 1911, D. Riabouchinsky published a paper which rediscovered Vaschy’s results.12 Riabouchinsky made this discovery while analyzing data he had generated at the Aerodynamic Institute of Kutchino. He provided a mathematical foundation for Dimensional Analysis and he stated, as did Vaschy, Buckingham’s Pi theorem. Riabouchinsky apparently rediscovered the foundation of Dimensional Analysis independent of Vaschy.

In 1914, Richard Tolman and Edgar Buckingham each published an article concerning Dimensional Analysis in the Physical Review.13,14 Tolman used Dimensional Analysis to investigate Debye’s recently published theory of specific heat. Buckingham investigated the foundation of Dimensional...