A History of Units and Dimensional Analysis
John Schulman
March 17, 2010
I.
INTRODUCTION
This paper is not primarily concerned with the development of any branch of physics but
rather with the language used by physicists in their derivations and results. Galileo (1564 1642), who is arguably the father of mechanics, wrote
If two particles are carried with uniform motion, but each with a different speed,
the distances covered by them during unequal intervals of time bear to each
other the compound ratio of the speeds and time intervals.
A modern scientist would write d = vt, meaning distance equals velocity times time.
In the works of Galileo, Huygens, and Newton, one never finds a numerical value for a
velocity, an acceleration, or a force. Instead, results are stated in terms of ratios (as in
the above quotation). Any calculation that one could perform in the modern way, using
equations that relate dimensioned quantities, could conceivably be performed Galileo’s way,
using ratios. However, the modern way has significant advantages: it allows for conversion
between different systems of measurement, e.g., meters and feet; it is much more concise;
and it enables dimensional analysis, where one infers the form of a physical equation without
doing any physics.
This paper chronicles the development of the idea of units and dimensions in physics.
There was a gradual transition from Galileo’s ratios to equations that relate dimensioned
quantities. The first man to have a modern understanding of dimensioned quantities was
Fourier, who stated these principles in the later editions of The Analytic Theory of Heat
[1]. We briefly discuss some of the advances that occurred after Fourier, namely the technique of dimensional analysis, where one infers the form of a equation using dimensional
considerations on the quantities involved.
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II.
HOMOGENEITY FROM EUCLID TO DESCARTES
The principle of homogeneity states that if A = B is physically meaningful, then A must
have the same units as B. Similarly, if A + B appears in an equation, then A must have
the same units as B. For example, it does not make sense to add a length to an area. An
early statement of this principle can be found in Euclid’s Elements, where Euclid states that
“only things of the same kind can be compared to each other,” where angles, lengths, areas,
and volumes are different kinds [2].
Greek mathematics was based on geometry, and arithmetic expressions were considered
to be sums and products of lengths and areas–everything had a geometrical interpretation.
Pappus (c. 290 - c. 350 AD) criticized proofs that contained a product of four lengths,
which did not have a physical interpretation [3]:
There are only three dimensions in geometry, although certain recent writers
have allowed them to speak of a rectangle multiplied by a square without giving
any intelligible idea of what they meant by such a thing.
Descartes broke this Greek tradition when he developed analytic geometry. He considered
curves in the plane described by equations like y = x3 −3x2 . Assume that x and y are lengths.
Descartes justified these equations by stating that the higher-degree parts (x2 and x3 ) are
implicitly multiplied by some power of the unit length u, so that the equation actually reads
y = x3 /u2 − 3x2 /u and all terms have the dimension of length. This means that analytic
geometry requires a special reference length, the unit length, whereas in Euclidean geometry
there is no special length.
III.
GALILEO AND THE GEOMETRY OF MOTION
Galileo’s Two New Sciences [4] is mostly concerned with kinematics: describing trajectories and travel times of objects moving without air resistance. Galileo speaks of the “speed”
of particles, but his results are stated in terms of ratios of distances and times.
Recall the quotation from the introduction:
If two particles are carried with uniform motion, but each with a different speed,
the distances covered by them during unequal intervals of time bear to each
other the compound ratio of the speeds and time intervals.
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(Proposition IV, Third Day). Galileo did not think it was sensible to take the product of a
speed and a time as we would in the modern statement of this theorem: d = vt.
Galileo never speaks of numerical values of speeds, though he refers to ratios of speeds of
non-accelerated objects and average speeds of accelerated objects (though he does not call it
this; he talks about an equivalent object that undergoes uniform, non-accelerated motion.)
The following passage sheds light on his thoughts:
Concerning motions and their velocities or momenta, whether uniform or naturally accelerated, one cannot speak definitely unless he has established a measure
for such velocities and also for time. As for time we have already widely adopted
hours, first minutes, and second minutes. So for velocities, just as for intervals
of time, there is need of a common standard which shall be understood and
accepted by everyone, and which shall be the same for all. . .
Let us consider the speed and momentum acquired by a body falling through
the height, say, of a spear, as a standard which we may use in the measurements
of other speeds and momenta as occasion demands.
These deliberations about a proper unit for speed sound peculiar to our modern ears–why
doesn’t he consider the length unit over the time unit? First, distance per time has at least
two possible meanings: the distance traveled by an object uniformly accelerated from rest
and the distance traveled by an non-accelerated object with constant speed. Galileo uses
the former meaning in his definition of velocity, possibly because it is a much easier way
of reliably producing a given speed. While Galileo seems to have a basic understanding of
the instantaneous velocity of an accelerated object (see below), this concept was not formal
and intuitive before the development of the calculus. Second, the idea of dividing units was
strange and counterintuitive to people at the time of Galileo. After years of practice, a
modern science student takes it for granted that the objects he manipulates in physics are
not the numbers of arithmetic. In the parlance of mathematicians, they are elements of a
graded ring: R[m, m−1 , kg, kg−1 , s, s−1 ] ([5]). Third, there were several systems for measuring
distance, none of which were standardized, so one may speculate that scientists preferred to
write of ratios of lengths rather than lengths to make their results more reproducible.
The later parts of Galileo’s Two New Sciences contain derivations of some rather nontrivial results, many of which would be challenging for a modern physics student to derive.
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The modern approach would involve algebraic manipulation of dimensioned quantities, but
Galileo (and Newton) had a different approach: they drew geometric diagrams where different points represented positions, velocities, and times.
Let the motion take place along the line ab, starting from rest at a, and in this
line choose any point c. Let ac represent the time, or the measure of the time,
required for the body to fall through space ac, let ac also represent the velocity
at c acquired by a fall through the distance ac.
(Proposition III, Fourth Day)
Evidently, the length ac is being used as a length, a velocity, and a time! Galileo is
effectively choosing units for length and time so that d = gt2 /2 = gt = t, i.e. units where
√
g = 1 and t = 2. Then he derives, for example, that a certain point s (between c and b)
is the velocity of the object when its position is b.
In the next section, Galileo uses these geometric techniques to derive various properties
of parabolic trajectories. Problem 1, Proposition IV is “To determine the momentum of a
projectile at each point in its given parabolic path”. From this we see that he did have a
concept of an instantaneous velocity.
In his Principia Mathematica [6] (1687), Newton (1643 - 1727) talks of forces and other
dimensioned quantities. Newton treats forces in a similar way to how Galileo treated velocities: he acknowledges that force has a physical meaning, and he talks about ratios of forces,
but he never talks about the numerical value of a force, and he never writes an equation
where force appears alone on one side. The modern statement of Newton’s second law is is
F = ma, but Newton wrote
A change in motion is proportional to the motive force impressed and takes place
along the straight line in which that force is impressed.
In the section on the contribution of the sun and moon to tides:
Wherefore since the force of the sun is to the force of gravity as 1 to 12868200,
the moon’s force will be to the force of gravity as 1 to 2981400
(Proposition XXXVII, PROBLEM XVIII, BOOK 3)
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IV.
THE METRIC SYSTEM
In the 1790s, French scientists debated about official standards for measurements, and in
1795 the metric system became the official system of measurement in France. The second
was a conventional unit for time, but the meter and kilogram were completely novel. These
units were designed so that any laboratory could perform measurements to reproduce the
standard lengths, masses, and other units. Previous measurement systems were standardized
by fixed prototypes.
The original definitions of units were as follows [7]:
• meter: 1/10,000,000 the distance between the North Pole and the equator
• second: 1/86,400 of the mean solar day
• kilogram: the mass of a liter of pure water, i.e., a cube with side 1/10 meter.
• The Celsius temperature scale was based on the properties of water, where 0 is freezing
point and 100 is the boiling point at atmospheric pressure
While it took a long time for these units to be officially adopted by most governments
(the Treaty of the Meter was in 1875), these units were quickly adopted by scientists and
engineers. For example, an 1804 article in the English journal Philosophical Magazine describes an method to measure the initial speed of a cannonball shot out of a cannon [8]. The
article gives an equation for the speed in terms of several measured quantities and lists the
results in meters per second.
Did standardized units affect the discourse of physics, enabling physicists to write equations involving dimensioned quantities, rather than Galileo and Newton’s ratios? This proposition is debatable. Euler’s physics, such as his work on ship stability [9], originally published
in 1749, is full of equations, some of which contain dimensioned constants like the gravitational acceleration g. So physics was written as equations before the metric system. But the
metric system certainly made this format more convenient, since one did not have to worry
that he was measuring distance in Paris feet while using the value of g measured in English
feet.
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V.
FOURIER AND THE NATURE OF PHYSICAL QUANTITIES
Josef Fourier (1768-1830) was the first to recognize that physical quantities have units
associated with them and that this determines how they rescale under changes in units. He
presented his new ideas in a chapter “General Remarks” of The Analytic Theory of Heat
(Section IX). There he states that physical quantities have dimensions:
In order to measure these quantities and express them numerically, they must
be compared with different kinds of units, five in number, namely, the unit of
length, the unit of time, that of temperature, that of weight, and finally the unit
which serves to measure quantities of heat.
Fourier’s calculations involve three constants which differ between materials and have a
non-obvious dependence on the fundamental units: specific heat capacity, thermal transmittance, and thermal conductance. Fourier’s calculations should hold regardless of the
measurement units (e.g., switching from meters to feet.) How do these three constants
change when the measurement units are changed?
Fourier realized that these constants have a dependence on the units of measurements,
and they must rescale when the measurement units change. The proper rescaling factor
depends on the exponent of dimension. For example, an area and a volume have exponent
of dimension 2 and 3 with respect to the unit of length. The specific heat c, heat to raise a
volume by 1◦ C, has exponent of dimension -3 with respect to length, so if the unit of length
is multipled by λ, then c → cλ−3 . Fourier gives a table specifying the dimensions of the
undetermined constants (modified here to use modern names for quantities):
Quantity or constant
Length Duration Temperature
Distance
1
0
0
Time
0
1
0
Temperature
0
0
1
Thermal conductance
-1
-1
-1
Thermal transmittance
-2
-1
-1
Specific heat capacity
-3
0
1
Fourier points out a generalized principle of homogeneity (generalized from length to
other types of units), which applies to equations and differential equations:
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On applying the preceding rule to the different equations and their tranformations, it will be found that they are homogenous with respect to each kind of
unit, and that the dimension of every angular or exponential quantity is nothing.
In other words, just as we cannot add a length to an area, we also cannot add quantities
A and B if A is per second and B is per second squared. Any quantity in an exponent must
be dimensionless.
Fourier’s insights on dimensional analysis far surpassed those of anyone who preceded
him. He did not realize, however, that the dimensions of the quantities involved could be
used to infer the form of the equation relating them–this technique would be used 50 years
later by Lord Raleigh.
VI.
THE RISE OF DIMENSIONAL ARGUMENTS
By describing physics with units, we can use the powerful technique of dimensional analysis. We can infer what a physical relation will look like just by knowing what quantities are
involved–not using any physical arguments. According to Martins [10], the first published
dimensional argument is due to Foncenix (1734 - 1799), published in 1761. Foncenix considers the total force R that results when two forces with equal magnitude F and an angle
A between them are applied to the same object. R = f (F, A) for some function f . Since
R and F have the same units but A is dimensionless, the relation must take the restricted
form R = F · g(A) for some function g. While this may seem like a trivial result, it was
quite innovative for its time, when no one talked about the dimensions of force. This paper
of Foncenix was cited by Legendre, Poisson, and Fourier.
The first man to productively use dimensional arguments was Lord Rayleigh (1842 - 1919),
who extensively used them in his Theory of Sound (1877, 1888) [11] and other papers, and
the Theory of Sound includes a section titled “Method of Dimensions.” A simple example
of dimensional analysis (similar to an argument used by Rayleigh) is as follows. Problem:
find the frequency of the small oscillations of a mass at the end of a massless spring. The
frequency f is a function of the mass m, the spring constant k, and the amplitude of the
oscillations A
T = g(m, k, A)
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(1)
The units of these quantities are as follows, where M, L, T are mass, length, and time,
respectively:
Variable
Dimension
f
T −1
m
M
k
M T −2
A
L
g(m, k, A) must have dimension T −1 . Raleigh’s argument was that the relation f =
g(m, k, A) should hold regardless of the choice of units; thus both sides must rescale in the
same way when we change units.
Then we must find a, b, c such that
M a (M T −2 )b Lc = T −1
(2)
This turns into a set of linear equations that must be solved, giving the unique solution
a = −1/2, b = 1/2, c = 0.
r
f = Const · k 1/2 m−1/2 = Const ·
k
m
(3)
(Note that there is no dependence on A.) Dimensional analysis will not tell us the constant,
but the exact formula is
1
f=
2π
r
k
m
(4)
In the 1880s and 1890s, a series of French scientists began to develop a more general and
rigorous theory of dimensional analysis. Vaschy wrote in 1896
Any homogenous relationship among p quantities a1 , a2 , · · · , ap , the values of
which depend on the choice of the units, may be reduced to a relationship involving (p − k) parameters that are monomial combinations of a1 , a2 , · · · , ap and
that are of zero dimensions (L0 M 0 T 0 = 1) [. . . where k ≤ n, the number of
fundamental units]
Vaschy’s powerful theorem was forgotten. It was rediscovered, probably independently,
by Riabouchinsky of of Russia in 1911. Edgar Buckingham wrote a series of papers on the
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subject in 1914, citing Riabouchinsky, and the Vaschy’s theorem above is now commonly
called the Buckingham π-theorem.
VII.
CONCLUSION
There are several threads in the history of units and dimensional analysis. First, the
principle of homogeneity (i.e., do not add lengths to areas) was generalized from lengths to
the other fundamental units including time and mass. Fourier realized that each quantity
has exponents of dimension with respect to the fundamental units (length, time, etc.) and
that both sides of an equation must have the same exponents. Second, the theory of real
analysis, including the concept of differentiation, was developed. This theory relates to the
adoption of compound units like velocity, since to measure velocity in meters per second, one
needs to realize that even accelerated motion has an instantaneous velocity, corresponding to
the time-derivative of position. Third, standardized units were developed–the metric system.
Fourth, the technique of dimensional analysis–using dimensional arguments to infer the form
of a physical relation–was developed by Raleigh and then formalized by other scientists.
The modern language for describing physics, equations involving dimensioned quantities,
is a more concise and illuminating way to write physics than Gallileo’s style, which relied
on ratios and made heavy use of plane geometry. The modern language makes possible
the powerful technique of dimensional analysis. Dimensional analysis was not discovered
until nearly 250 years after Galileo’s Two New Sciences. How can this history guide us in
developing a language to discuss physics? First, one should be able to write down physics
in a way that does not contain arbitrary choices (i.e., the choice of measurement units).
However, the language should not disallow us from making these arbitrary choices; it must
provide a way to translate between them. When you write d = 21 gt2 , where g = 9.8m/s2 ,
you have made an arbitrary choice of measurement units (meters and seconds), but I know
exactly how to translate between your units and my units (e.g., miles and hours). Second,
assign a mathematical object to anything that has physical meaning. Galileo knew that
velocity had physical meaning, and Newton knew that forces had physical meaning, but
they did not assign them to mathematical objects that could stand alone. Now, a physicist
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is just as comfortable writing
kg·m/s2
(for force) as he is writing down m (for length).
[1] J. B. J. Fourier. The analytical theory of heat. Cambridge University Press, 1822.
[2] S. Rudolph. On the foundations and applications of similarity theory to case-based reasoning.
In Proceedings of the Twelvth International Conference for Applications of Artificial Intelligence in Engineering. Capri, Italy. Citeseer, 1997.
[3] Enzo O. Macagno. Historico-critical review of dimensional analysis. Journal of the Franklin
Institute, 292(6):391–402, December 1971.
[4] G. Galilei, H. Crew, A. Favaro, and A. de Salvio. Dialogues concerning two new sciences.
Dover Publications, 1954.
[5] A. Borovik. Mathematics under the microscope. Creative Commons, 2007.
[6] I. Newton, A. Motte, and N. W. Chittenden. Newton’s Principia: the mathematical principles
of natural philosophy. D. Adee, 1848.
[7] Robert A Nelson. The international system of units: Its history and use in science and
industry. Via Satellite, 15:90–109, 2000.
[8] Unknown. XXXIV. Report on the means of measuring the initial velocity of a projectile.
Philosophical Magazine Series 1, 22:220–231, 1804.
[9] Leonhard Euler. A complete theory of the construction and properties of vessels: with practical
conclusions for the management of ships, made easy to navigators. Translated by Henry
Watson, printed for J. Sewell, 1790.
[10] Roberto De A. Martins. The origin of dimensional analysis. Journal of the Franklin Institute,
311(5):331–337, May 1981.
[11] J.W.S. Rayleigh. The theory of sound: in two volumes. Macmillan, 1896.
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