Stevinus epitaph Thermodynamics meets number theory.pdf

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International Journal of Mathematical Education inScience and Technology

ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage: http://www.tandfonline.com/loi/tmes20

Stevinus’ epitaph: Thermodynamics meets numbertheory

Michael A.B. Deakin & G. J. Troup

To cite this article: Michael A.B. Deakin & G. J. Troup (1976) Stevinus’ epitaph: Thermodynamics

meets number theory, International Journal of Mathematical Education in Science andTechnology, 7:3, 271-276, DOI: 10.1080/0020739760070303

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INT. J. MATH. EDUC. SCI. TECHNOL., 1976, VOL. 7, NO. 3, 271-276

Stevinus ' epitaph:

Th e rm o d y n a m ic s m e e t s n u m b e r t h e o ry

by MICHAEL A. B. DEAKIN

Mathematics Department, Monash University, Australia

and G. J . TROUP

Physics Department, Monash University, Australia

(Received 9 March 1976)

A number of versions of the diagram known as the ' E pitaph of Stevinus '

are presented. The diagram has the purpose of demonstrating the law of

the resolution of forces from considerations of the impossibility of perpetual

mo tion. T he usual versions of the diagram require a number-theoretical

result, proved herein, for completeness of the account.

Simon Stevin (Stev inus), 1548-1620, produced the d iagram of figure 1 to

prove his law of resolution of forces. In this version, the figure appears as the

frontispiece of his Hypomnemata Mathematica of 1608, over the legend

Wonder en is gheen wonder , rendered by J. R. Maddox, translator of Dugas'

History of

Mechanics [1], as The Magic is not magical . Thi s, according to

Dugas, is Stevin's indication that he had logically explained a fact instead of

invoking magic as the Greeks had done, before Archimedes, in connection with

levers.

Figure 1. Th e Epitaph of Stev inus: Stevin's first version of the diagram. Redrawn from

Dugas [1]. Th e (erroneously) circular arc AC is a feature of the original, but the

precise form is irrelevant.

The salient features of figure 1 are the lengths of the lines BC, AB, which

are in the ratio 2 : 1 . T he four spheres or cylinders on the gentler slope AB

balance the two resting on BC. T he eight others form, by symm etry, a sub -

system in equ ilibrium. T he precise role of the points S, T, V will be discussed

later.

Stevinus then generalized this result to

WAB WBC

m

W fee

{)

D

o w n l o a d e d b y [ U n i v e r s i t ä t O s

n a b r u e c k ] a t 0 0 : 1 6 1 0 M a r c h 2 0 1 6



7

M.A.B. Deakin and G. J. Troup

where WAB,

W-RC

are the weights resting on the sides AB, BC, respectively, and

J B fee are the corresponding lengths. It is unde rstood in this that AC is to

be horizontal. Figure 2 gives a special, bu t imp ortan t case, and figure 3

indicates the more general situation.

Figure 2. A later version by Stevin; AB = 2BC. Redrawn from Du gas [1].

Figure 3. A generalized version of the Epitaph. AB : BC mu st be rational. AC m ust

be horizontal. Th e case shown here h as AB : BC = 3 : 4. The precise form of

the polygonal arc AC is not relevant.

Figure 2 is rather more stylized than figure 1 in that the continuous belt

of weights has been replaced by a configuration that is seen to be equivalent

once equation (1) is reached. Nevertheless, that continuous belt is required

for the general derivation of equation (1). T he propo rtionality of W, I on both

sides of the point B is assured by the construction. T he argum ent that

produces the static equilibrium is based on the impossibility of perpe tual motion.

W ere the re a tendency for the loop to rotate clockwise (say), this would

persist, and be due to an excess in the vertical component of

WBC

over the

vertical component of WAB- That these vertical components are equal leads

us to suspect that the vertical component is actually proportional to PFAB- T O

construct a proof along these lines it is preferable to employ a specialized form

of the diagram.

Figure 4 gives such a specialization. T his particular form is that reprod uced

in

The F eynman Lectures on Physics

[2] (which incidently repeats the story that

the diagram is carved on Stevin's tomb; the diagram and the motto Wonder

Figure 4. A specialization of figure 3. He re AB is vertical, AC horizontal. AB : BC

mu st be rational. Th e case shown here is A B : B C = 3: 5. Redrawn from

Feynman [2 ]. The form of the arc AC , which is in fact erroneous, is irrelevant.

D o w n l o a d e d b y [ U n i v e r s i t ä t O s n a b r u e c k ] a t 0 0 : 1 6 1 0 M a r c h 2 0 1 6



Stevinus' epitaph 273

en is gheen wonder are both referred to as The Epitaph of Stevinus. Either

would be a fitting monument, but we are unable to discover whether the story

is accurate or apocryphal).

An even simpler form is figure 5, where the slack has been taken u p. It

may also be noted that this version may be presented with the triangle inverted.

Figures 4 and 5 show the situation in which AB is vertical. WAB itself may

now be assumed to be the vertical component of W AB, SO tha t now we have

vertical component of W B — W B = vertical component of WBC (2)

But

WAB

= IABWBCIIBC = W

BC

cos 6 (3)

T hus

vertical

component

of

W ^c = W-QG COS

6

(4)

which is, essentially, the triangle law of force. A complete proof may readily

be constructed along these lines.

Figure 5. A more compact form of figure 4. In this version, the arc AC has collapsed

to a straight line and the triangle ABC could be inverted.

A number of objections may be raised against this derivation. In the first

place, the possible role of friction needs discussion. W e may discuss it to

dismiss it. Even withou t a precise law of friction, we may imagine tha t th e

friction plays a relatively smaller role as the weights WAB, WBC are increased

(by, for example, increasing the density of the spheres or cylinders represented

by the circles). Th is is particularly the case in the configurations of figures

4 and 5. W e may also imagine friction reduced by lubrication or use of bearings.

Friction, moreover, prevents motion in either direction; we can say that

equation (4) represents a central point of that set of possible equilibria, and is at

least approximately true.

But there is a more fundamental reason for discounting friction, because

ultimately we find the idea that the device generates perpetual motion repugnant,

even in a frictionless world. W e should not need to invoke friction to prevent

perpetual motion.

A more difficult objection is the question of whether or not the device might

catch or jam . Fey nm an's version, whose geome try we reprod uce exactly in

figure

4,

seems to be on the point of doing just this, and the mechanism of figure

5 would undoubtedly jam in many configurations even if there were a genuine

tendency to move—i.e. it would equilibrate even if the weights disobeyed

equation (1). Th is objection seems to have been foreseen by Stevin, who

inserts (see figure 1) the points S, T, V, three fixed points on which the cord

can run freely without being caught .

Quite how Stevin intended his diagram to be interpreted seems now to be

an unanswerable question. A num ber of quite feasible possibilities exist.

D o

w n l o a d e d b y [ U n i v e r s i t ä t O s n a b r u e c k ] a t 0 0 : 1 6 1 0 M a r c h 2

0 1 6



7

M. A. B. Deakin and G. J. Troup

One possibility is that the individual cylinders have the cross-section shown in

figure 6, the central axle passing through the cord arid the lower rims having

rolling or sliding contact with the surface of a triangular prism.

Figure 6. A possible cross-section for one of Stevin's cylinders. The line of sight is in

the plane ABC from the left of figure 1.

W e ourselves prefer to redraw the diagram slightly. Figu re 5 then takes on

the appearance shown in figure 7. T he attachment to the cord, or belt, is shown

in figure 8. Roller bearings may be inserted directly at the points A, B, C, any

resulting reduction in the lengths of the sides being reducible to arbitrarily

negligible prop ortions. Th ese roller bearings take the place of Stevin 's poin ts

S, T, V.

Figure 7. An alternative version of figure 5. The altered geometry is provided as an

answer to possible jamm ing.

Figure 8. A method of attachm ent to the cord or belt of figure 7.

A third possibility is to abolish the cylinders entirely and imagine a heavy

(and, given tha t this implies thickness, flexible) band as in figure 9. Th is too

leads to equations (1) and (4) and does so in a reasonably direct way. How ever,

this does not have the immediate impact of the other versions, such as figure 7.

A final difficulty does not arise with figure 9, but is relevant to the discrete

versions of the diagram. All such versions require lengths in integral ratios

if we are to apply the argument of equations (1); these ratios must also form a

right-angled triangle if the argument of equations (2)-(4) is to hold.

D o

w n l o a d e d b y [ U n i v e r s i t ä t O s n a b r u e c k ] a t 0 0 : 1 6 1 0 M a r c h 2

0 1 6



Stevinus epitaph

75

It is well known that the sides of such a triangle must be in the ratio

luv : u

2

—

v

2

:

u

2

+ v

2

,

where

u, v (u> v)

are positive integers;

u

2

+ v

2

is the

hypotenuse (BC in figures 4, 5 and 7). Th us

luv u

2

v

2

...

C 0 S

^

=

~ 2 2 °

r

~2 2

It is apparent that not all angles 6 can be analysed in terms of the discrete

versions. How ever, we shall show tha t all angles 6 may be arbitrarily approxi-

mated by angles f> satisfying equation (5). Th is theorem is unlikely to be new,

although we are unable to find a reference to it; the proof moreover is relatively

straightforward if the details are tediou s. W e presen t an outline.

Fig ure 9. A cont inuo us version of figure 7.

Le t us agree to call the fractions

{Iuvj{u

2

+ v

2

)}, {(u

2

—

v

2

)l(u

2

+ v

2

)} ,

'Pytha gore an fractions '. W e then have:

Theorem :

Any real number x in

(0, 1)

may be approximated to arbitrary accuracy by a

Pythagorean fraction.

Proof:

{A )

It will be sufficient to show that any such

x

may be approximated to

arbitrary accuracy by a fraction of the form {(w

2

v

2

)j(u

2

+ v

2

)} .

(B ) As every real x is approximable to arb itrary accuracy by a rational nu mb er

it will suffice to show that, for all e (> 0), there exist u, v such that

P__\~

q 1

< e

(C ) To find such u, v form \/ {(q+p)l(q—p)} • T his is a real number approxi-

mable to arbitrary accuracy by rationals. Let u

n

be the numerator and v

n

th e

denominator of the wth in a sequence of rationals tending to the required limit.

Clearly u

n

> v

n

. Then

u

n

2

v

n

2

• pjq

as w->oo

(D )

Hence the result. T he stated result on the angle

0

follows immediately

by setting # = cos

9

and use of the continuity of the cosine function.

The need for this result does not arise in the continuous version (figure 9),

but figures 5 and 7 require such a result if equation (4) is to be deduced with

any generality from them . T his point is, if not new, insufficiently publicized.

Moreover, it gives a case of a number theoretical result, albeit a straightforward

one, with application to mechanics.




276

Stevinus' epitaph

The use of thermodynamical principles, especially through the proposal of

some perpetual mo tion machine and the requirem ent that it fails, has, we suspect,

wide application in the derivation of physical laws. T he presen t paper deals

with a previously known but historically and mathematically interesting case.

We hope in later papers to explore other situations in the same spirit.

Acknowledgment

We thank Jean Hoyle, who drew the diagrams.

References

[1]

DUGAS,

R., 1957, A History of Mechanics, translated by J. R . M addox (Londo n:

Routledge & Kegan Paul Ltd .), pp. 123-127.

[2]

FEYNMAN,

R.,

LEIGHTON,

R., and

SANDS,

M., 1963,

The Feyn man Le ctures on Physics,

Vol. 1 (Reading, Massachusetts: Addison-Wesley), pp. 4 and 5.


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Stevinus epitaph Thermodynamics meets number theory.pdf