Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics

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3.3 History 25

Exercises

The following three exercises are from historic sources, they were kindly

sent to me by Albrecht Heeffer as a comment on the problem discussed in

this chapter:

It takes ﬁve days for a steamboat to get from St Louis to New

Orleans, and seven days to return back from New Orleans to St

Louis. How long will it take f or a raft to drift from St Loui s to New

Orleans?

Exercise 3.1 Albrecht Heeffer wrote to me:

While meeting and overtaking problems are abundant in Renais-

sance and early modern books on ari thmetic and algebra, prob-

lems involving a current or a wind appear rather late. The earliest

occurrence of such problems I could ﬁnd is from the end of the

eighteenth century:

“If during the T ide of Ebb, a Wherry should set out from

London westward, and at the same instant another should

put off at Chertsey for London, ta king the distance by wa-

ter at 34 miles : The stream forwards one and retards the

other, say 2

Miles an Hour : The Boats are equally l aden,

the Rowers equally good, and in the ordinary Way of work-

ing, i n still Water, would proceed at the ra te of 5 Mil es an

Hour : The Question is, where in the River the two Boats

would meet ?” [From The Tutors Guide: Being a Complete

System of Arithmetic, by Charles V yse, 1772, p. 211.]

Solve this problem.

Exercise 3.2 Albrecht Heeffer further comments that

Such problems are usually dealt with in a chapter on two equa-

tions with two unknowns:

Horatio N. Robin son. New Elementary Algebra: Contain-

ing the Rudiments of the Science for Schools and Academies.

Ivison, Blakeman, Taylor & Co., New York, 1875. Prob. 90,

p. 305.

“A person residing on the bank of the Ohio, 15 miles

above Cincinnati, can row his boat to the city in 2

hours,

but it requires 7

hours to return. With what force can he

row his boat in still water, and wh at is the velocity of the

river ?”

Exercise 3.3 And here is Albrecht Heeffer’s ﬁnal remark:

Although in your problem the dis tance is not being given, th e an-

swer can easily be given, if you know how to calculate the speed of

the current or tide. T his is nicely explained in the following book,

and the earliest occurrence of this kind of problems I could ﬁnd is

in Algebra Made Easy : Chieﬂy Intended for the Use of Schools, by

Thomas Ta te (London : Longman, Brown, Green, and Longmans,

1847), p. 86, no. 11.

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26 3 Uni ts of measurement

“A rower goes a mph with the tide and b mph against the

tide. What is the rate of the tide?”

Again, solve it.

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History of Dimensional Analysis

Der wür de nicht gelebt ha ben,

der nur in der Gegen wart lebte.

Heinrich Mann

It is time to f ormulate the ke y principle of dimensional analysis:

two physical quantities have the same dimensions if their measure-

ments change in exactly the same way under all possible choices

of units of measurement. For example, the v alue of acceleration is

more sensitive than velocity to a change of units of time, but be-

haves similarly to velocity under a change of units of length. But,

instead of change of units, we can vary the size and other param-

eters of o ur o bjects and processes—then the same scaling consid-

erations will give us important insights into their be havior. In our

ﬁrst example, it makes sense to stay strictly within mathematics.

A bit later we shall turn won derful applications of the scaling prin-

ciples to physical laws. We start with the basic observ ation: area

of geometric ﬁgure changes as squares of its characteristic linear

measurement. For example, if we increase a side of a square by

factor of two, its area increases by factor of four. This observation

allows us to p rove Pythagoras’ Theorem:

If a and b are si d es and c the hypothenuse of a right-angled trian-

gle, then

+ b

= c

Indeed , consider all triangles similar to the given triang le with

sides a and b and the hypo tenuse c; the correspond ing sides a

′

, b

′

and the hypothenuse c

′

are obtained from a, b, c by stretching with

the same c oefﬁcient. We can take the hypote nuse c for character-

istic linear measurements, then area S of the triangle changes as

square of c, that is,

S = kc

for some coefﬁcient k which is the same for all similar triangles.

The key step now is that the right-angled triangle can be cut in

28 4 History of Dimensional Analysis

two similar triangles in which a and c become hypotenuse s, and the

area S becomes the sum of the areas of the two smaller triangles,

= ka

+ kb

which instantly gives us Pythagoras’ Theorem:

= a

+ b

This argument apparently has a long history. Alexander Given-

tal [186] comments:

In fa ct the foregoing argument is not new. Moreover, it is so close

to the original proof of [Euclid’s] Proposition VI.31 that G. Polya

[75] attributes it to Euclid himself.

At this point, we turn to som e wo nderful examples from the

history of ph ysics an d the natural sciences.

4.1 Galileo Galilei and the ﬁrst of his two “ New

Sciences”

Dimensional analysis has a fascinating history indeed. Its origins

can be trace d back to the seminal book by Galileo Galilei Two New

Sciences [184]. In a modern day assessment (Peterson [218]),

Galileo’s last book was the Two New Sciences, a dialogue i n four

days. The third and fourth days describe his solution to the long-

standing problem of projectile motion, a result of obvious impor-

tance and the birth of physics as we know it. But this was only the

second of his two new sciences. What was the ﬁrst one?

Two New Scien ces begins in the Venetian Arsenal, the shipyard of

the Republic of Venice, with a discussion of the effect of scaling up

or scal ing down in practical constructi on projects, l ike shipbuild-

ing. [. . . ] According to the publis her’s foreword, it is this topic that

should be understood as the ﬁrst of the two n ew sciences.

Galileo’s observations on scaling in general are i ngenious a nd ele-

gant, and entirely deserving of the prominent pla ce he gives them.

These ideas are basi c i n physics, and are introduced in most i ntro-

ductory physics texts under the heading of dimensional analysis.

We could even say that modern renormalization group methods

are just our most recent way to deal with problems of scale, still

recognizably in a tradition p ioneered by Galileo.

Just re ad this unbelievable quote from Galileo, a formulation of

the scaling pr oblem by one of the discussants in the bo ok, Sagredo :

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4.1 Gali leo Galilei 29

“Now, because mechanics has its foundation in geometry, where

mere size cuts no ﬁgure, I do not see that the properties of cir-

cles, triangles, cylinders, cones and other solid ﬁ gures will change

with their size. If therefore a large machine be constructed in such

a way that its parts bear to one another the sa me ratio as in a

smaller one, and if the smaller is sufﬁciently strong for the pur-

pose for which it was d es igned, I do not see why the larger also

should not be [sufﬁciently strong] . . . ”

Sagredo clearly formulates the principle that geometric proper-

ties are invariant under scaling transformations, and he wants to

understand why this invariance is broken in the real world. The

answer is proposed by Salviati:

“. . . these forces, resistances, moments, ﬁgures, etc. may be consid-

ered in the abstract, dissociated from matter, or in the concrete,

associated with matter. Hence the properties which belong to ﬁg-

ures th at are merely geometrical and nonmaterial must be modi-

ﬁed when we ﬁll these ﬁgu res with matter a nd therefore give them

weight.”

In modern parlance, change of pro perties results in a change

of the group of permitted transfor mations and breaks scale in vari-

ance, but the new theory still allows a development in terms of

invariants. Salviati continues:

“Since I assume matter to be unchangeable and always the same,

it is clear that we are no less able to treat this constant and invari-

able property in a rigid manner than if it belonged to simple and

pure mathematics.”

Galileo then uses these general principles, for example, to e x-

plain why smaller objects fall more slowly than big ones: their

area varies as the square of linear dimension, while their weight

varies as the cube of linear dimension, and therefore for smaller

objects surface forces—such as air drag—become more signiﬁcant

compared to the force of weig ht.

He conside rs ho w ﬁbrous materials, like wood beams, break un-

der their weight an d makes a now famous rem ark that if we scale

an animal up, its bones should become thicker and thicker in com-

parison with their length. We shall return to the discussion of this

in the exercises.

The reader who wants to learn more about how Galileo’s ideas

continue to live in ph ysics can ﬁnd a concise and clear outline of

the modern understanding o f dimensional analysis in a beautiful

little book by Yuri Manin [59].

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30 4 History of Dimensional Analysis

And I conclude this brie f introd uction to principles of dimen-

sionality by a childhood testimony from my friend Hovik Khudav-

erdyan

I was about 10 when my parents bought me as gift a scaled plastic

model of Ilyushin IL-18 plane; it was made to 1:100 scale and its

weight was, according to the la bel, 154 gram. I was as king my

parents: how could this happen, the empty weight of IL-18 was

35 tons, 1/100 of it had to be 350 kilograms? Even if plastic was

lighter than metal, this could not explain the discrepancy!

4.2 Froude’s Law of Steamship Comparisons

The next signiﬁcant step in the developm ent of dimensional analy-

sis appears to be Froude’s Law of Steamship Comparisons:

the maximal speed of similarly designed steamships is propor-

tional to the square root of their length .

William Froude ( 1810–1879) was the ﬁrst to formulate reliable

laws for the resistance that water offers to ships and for predict-

ing their stability. In this section, we give a deduction of this law

adapted from D’Arcy Thompson [840, p. 24]. But ﬁrst we have to

discuss some difﬁculties of the mathematical modeling of physical

phenomena and the limitations of dimensional analy sis.

4.2.1 Difﬁculty of making physical models

We need to understand ﬁrst how the drag F (force of resistance)

offered by water to a ship depends on the speed of the ship. Sur-

prisingly, it is is easier to do for high speeds than for lower ones. At

high speeds we can assume that the drag is offered by water being

violently thrown away from the course of the ship, ig noring a ﬁner

picture of what is happenin g with the water. Obviously, the drag

F sho uld depend on the crossection area S of the ship, its speed

V , and density of water ρ: the heavier the water, the harder it is to

throw it aside. Therefore we are looking for an equation in the form

F = cρ

with a dimensionless coefﬁcient c. Substituting dimensions, we

have

mass · length

time



mass

length





length





length

time



HK is male, Armenian, in itial mathematics education was in Armenian.

He is a physicist by education who moved to research in physics-inspired

geometry, holds a PhD, teaches in a British university.

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4.2 Froude’s Law of Steamship Comparisons 31

and immediately see that z = 2, x = y = 1, and the so-called For-

mula of Quadratic Drag takes the form

F = cρSV

According to Wikipedia, this equation is attributed to Lord Rayleigh.

At lo wer speeds, water ﬂows smoothly aro und the hull of the

ship, and a much more de licate analysis is needed. This has been

implemented by Sir George Stokes

(1819–1903) for the special

case of small spherical o bjects moving slowly through a viscous

ﬂuid. It is

F = −6πηr V,

where r is the effective radius of the object, η is the viscosity o f the

ﬂuid.

In short, for small objects like bacteria the dr ag is p r oportional

to their length and velocity; we shall denote this symbolically as

F ∼ LV.

Notice that the prop ortionality coefﬁcient is no long er dimension-

less.

Sadly, Stokes’s beautiful mathematical deduction does not apply

to real ships.

4.2.2 Deduction of Froude’s Law

Instead of Raleigh’s and Stokes’ drag formulae, we have to use the

property (and we shall treat it simply as an experimental fact, as

Froude did after a number of experiments) that, at small speeds,

the drag F is proportion al to the area of cro ss section of the ship

(and hence to the square L

of its typical linear size L) and to the

velocity V of the ship. We shall write this symbolically as

F ∼ L

and call it Froude’s drag.

To sustain constant speed, the ship has to pr oduce power F V .

Now w e have to recall that we are talking about steamships, pow-

ered by an engine inside, this engine working on steam produced

by burning coal, this coal being carried by the ship, etc. Therefore

Stokes is likely to be known to the reader as the author of the Stokes

Theorem for surface integrals:

dω =

∂M

ω.

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32 4 History of Dimensional Analysis

it is reasonable to assume that the power produc ed by the ship is

proportional to its volume, and therefor e

F V ∼ L

Combining the two proportions together, we have

∼ L

and there fore

V ∼ L

1/2

This is the promised Froude’s Law of Steamship Comparisons.

4.3 The triumph of “named numbers”:

Kolmogorov’s “5/3” Law

To demonstrate the power of “named numbers”, I borrow this sec-

tion from my book Mathematics un der the Microscope [103].

The deduction of Kolmogorov’ se minal “5/3” law for the en ergy

distribution in the turbulent ﬂuid [52] is so simple that it can be

done in a few lines. It remains the most striking and beautiful ex-

ample of d imensional analysis in mathe matics.

I was lu cky to study at a good secondary school where my

physics teacher (Anatoly Mikhailovich Trubachov, to whom I ex-

press my eternal gratitude) de rived the “5/3” law in one of his

improvised lectures. In my exposition, I borrow some details from

Arnold [96] and Ball [3] (where I also picked up the idea of using a

woodcu t by Katsushika Hokusai, Figure 4.1, as an illustration).

4.3.1 Turbulent ﬂow s: b asic setup

The turbulent ﬂow of a liquid con sists of vortices; the ﬂow in every

vortex is made of smaller vortices, all the way down the scale to

the point when the viscosity of the ﬂuid turns the kinetic energy of

motion into h eat (Figure 4.1). If there is no inﬂux of energy (like the

wind whipping up a storm in Hokusai’s woodcut), the energy of the

motion will eventually dissipate and the water will stand still. So,

assume that we have a balanced energy ﬂow, the storm is already

at full stren gth and stays that way. The motion of a liquid is made

of waves of different lengths; Kolmogorov asked the question, what

is the share of energy carried by waves of a particular length?

Here is a somewhat simpliﬁed description of his analysis. We

start by making a list of the quantities involved and their dimen-

sions.

First, we have the energy ﬂow (let me recall, in our setup it is

the same as the dissipation of energ y). The dime nsion of energy is

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4.3 Kolmogorov’s “5/3” Law 33

Fig. 4.1. Multiple scales in the motion of a ﬂuid, from a woodcut by Kat-

sushika Hokusai The Great Wave off Kan agawa (from the series Thirty-six

Views of Mount Fuji, 1823–29). This image is much beloved by chaos sci-

entists. Source: Wikimedia Commons. Public domain.

mass · length

time

(remember the formula K = mv

/2 for the kinetic ene rgy of a mov-

ing m aterial point). It will be convenient to make all calculations

per unit of m ass. Then the en ergy ﬂow ǫ has d im ension

energy

mass · time

length

time

For counting waves, it is convenient to use the wave number, that

is, the number of waves ﬁtting into the u nit of len gth. Therefo r e

the wave number k has dimension

length

Finally, the energy spectrum E(k) is the quantity such that, given

the interval

∆k = k

− k

between the two wave numbers, the energy ( per unit of mass) car-

ried by waves in this interval should be approximately equal to

E(k

)∆k. Hence the dimension of E is

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34 4 History of Dimensional Analysis

energy

mass · wave nu mber

length

time

4.3.2 Subtler analysis

To make the next crucial calculations, Kolmo gorov made the major

assumption that amounted to saying that

The way bigger vortices are made from smaller ones is the same

throughout th e range of wave numbers, from the biggest vortices

(say, like a cyclone covering the whole contin ent) to a smaller one

(like a whirl of dust on a street corner).

Then we can assume that the e nergy spectrum E, the energy

ﬂow ǫ and the wave number k are linked by an equation which

does not involve anything else. Since the three quantities involved

have completely different dime nsions, we can combine them only

by means of an equation of the form

E(k) ≈ Cǫ

· k

And now the all-important scaling considerations come into the

play. In the equation above, C is a constant. Since the equation

should remain the same for small scale and for global scale events,

the shape of the equation should not depend on the choice o f units

of measureme nts, hence the constant C should be dime nsionless.

Let us now check how the equation looks in terms of dimensions:

length

time

length

time



length



After equating lengths with le ngths and times with times, we have

length

= length

· length

−y

time

= time

which leads to a system of two simultaneous linear equations in x

and y,

3 = 2x − y

2 = 3x

This can be solved with ease and gives us

x =

and y = −

This formulation is a bit cruder than most experts would accept; I bor-

row it from Arnold [96].

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