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TYCHOMANCY
Inferring Probability from
Causal Structure
MICHAEL STREVENS
Harvard University Press
Cambridge, Massachusetts
London, England
2013

Copyright © 2013 by the President and Fellows of Harvard College
All rights reserved
Printed in the United States of America
Library of Congress Cataloging-in-Publication Data
Strevens, Michael.
Tychomancy : inferring probability from causal structure / Michael Strevens.
pages cm
Includes bibliographical references and index.
ISBN 978-0-674-07311-1 (alk. paper)
1. Probabilities. 2. Inference. 3. Empiricism. I. Title.
QC174.85.P76S74 2013
003'.1—dc23 2012042985

CONTENTS
Author’s Note xi
Physical Intuition 1
I
1859
1. The Apriorist 7
2. The Historical Way 13
3. The Logical Way 26
4. The Cognitive Way 38
II
Equidynamics
5. Stirring 53
6. Shaking 71
7. Bouncing 93
8. Unifying 113

Contents
viii
III
Beyond Physics
9. 1859 Again 127
10. Applied Bioequidynamics 149
11. Inaccuracy, Error, etc. 160
IV
Before and After
12. The Exogenous Zone 185
13. The Elements of Equidynamics 205
14. Prehistory and Meta-History 217
Notes 229
Glossary 245
References 249
Index 257

ix
FIGURES
1.1 Maxwell’s velocity distribution 8
2.1 Approach circle for molecular collision 19
3.1 Dodecahedral die 27
4.1 Ball in box with multiple exits 44
5.1 Evolution function for wheel of fortune 56
5.2 Different density, same probability 57
5.3 Nonmicroequiprobabilistic density 58
5.4 Evolution function for two consecutive spins 60
5.5 Evolution function for tossed coin 64
6.1 Bouncing coin landing conditions 72
6.2 Motley wheel of fortune 79
6.3 Motley wheel evolution function 79
6.4 Level one embedding trial 80
6.5 Level two embedding trial 81
7.1 Complex bouncers 109
11.1 Curve-fitting 164
11.2 Three possible error curves 165
12.1 Microequiprobability and rogue variables 187

xi
AUTHOR’S NOTE
A set of rules is presented in this book for inferring the physical probabil-
ities of outcomes from the causal or dynamic properties of the systems
that produce them—the rules of what I call equidynamics. The probabil-
ities revealed by the equidynamic rules are wide-ranging: they include
the probability of getting a 5 on a die roll, the probability distributions
found in statistical physics, the probabilities that underlie many prima
facie judgments about fitness in evolutionary biology, and more.
Three claims are made about the rules: that they are known, though
not fully consciously, to all human beings; that they have played a cru-
cial but unrecognized role in several major scientific innovations; and
that they are reliable. The arguments for these claims are, respectively,
psychological, historical, and philosophico-mathematical. Psychologists,
historians, philosophers, and probability theorists might therefore find
something interesting in the following pages, but they will have to find
their way through or around some rather foreign-looking material.
Perhaps the most challenging sections are the mathematical arguments.
As each of the rules of equidynamics is dredged up from the cognitive
depths and displayed in turn, I give some kind of explanation of the rule’s
ecological validity, that is, its reliability in the environments and contexts
in which it is normally put to use. Some of these justifications are reason-
ably complete, except for formal proofs. Some are mostly complete. For
some justifications you are referred elsewhere, although what’s elsewhere
is itself only partially complete. For some I provide just a hint as to how
a justification would proceed. In all cases except the last, the arguments
for ecological validity have a certain degree of mathematical complexity.
These passages are not easy. Please feel free to skip past them; they are

Author’s Note
xii
an important part of the complete argument for the theory of equidyn-
amics, but they can be coherently detached from the rest of my case.
It is less easy to skip the history or psychology, but if you are deter-
mined, you could read the book as an equidynamics manual with copious
illustrations drawn from historical and everyday thought.
Note that the book contains a glossary of technical terms.
My thanks go to Iowa State University for a seed grant to begin thinking
about Maxwell and equidynamics back in 1996, to Stanford University
for a year of junior leave used partially to draft a long paper about the
roots of Maxwell’s derivation of the velocity distribution, and to the Na-
tional Science Foundation for a scholar’s grant enabling a year of leave
from teaching and administrative responsibilities (2010–2011) during
which this book was written. Formal acknowledgment: This material is
based upon work supported by the National Science Foundation under
Grant No. 0956542.
Thanks also to Michael Friedman, Fred Kronz, and several anonymous
nsf reviewers for comments on the research proposal; to audiences at
Stanford University and a workshop on probabilistic models of cognitive
development held at the Banff International Research Station for com-
ments on talks presenting parts of the project; and to André Ariew, Laura
Franklin-Hall, Eric Raidl, and two anonymous Harvard University Press
readers for comments on the manuscript.

Well it was an even chance . . .
if my calculations are correct.
TOM STOPPARD,
Rosencrantz and Guildenstern Are Dead

1
P H Y S I C A L I N T U I T I O N
Do scientists know what they’re doing? Not always, not any more than
the rest of us. We may do it well, but often enough, we could not say
how—how we walk, how we talk, and in many respects, how we think.
Some elements of scientific procedure, such as classical statistical test-
ing or running randomized, controlled experiments, have their essentials
made explicit in textbooks or other canonical documents. Some ele-
ments, such as the management of various fussy and fragile experimental
setups, are typically left tacit, and are communicated from mentor to
scientific aspirant through hands-on, supervised learning-by-doing in the
lab or field or other practical venue. And some elements involve schemes
of inference or other forms of thought that do not seem to be learned at
all—or at least, there is no obvious period of apprenticeship in which a
student goes from being a novice to a master of the art.
The third class includes a rather heterogeneous cluster of abilities
called “physical intuition.” One part, perhaps the principal part, of physi-
cal intuition is the ability to inspect an everyday situation and, without
invoking theoretical principles or performing calculations, to “see” what
will happen next; you can, for example, see that putting pressure on the
rim of a round, three-legged table midway between any two legs may
unbalance the table, but putting pressure on the rim directly over a leg
will not (Feynman et al. 2006, 52–53). We humans all have this facility
with physical objects, this physical intuition, and we have a correspond-
ing ability to predict and understand without calculation the behavior
of organisms, minds, and groups—we have, you might say, biological,
psychological, and social intuition.
What we do not have is the ability to see how we see these things: the
principles guiding our intuitive physics are opaque to us. We “know more

Physical Intuition
2
than we can tell” (Polanyi 1961, 467). Or at least, than we can tell right
now, since presumably the sciences of the mind are capable of unearthing
the architectonics of intuition and thereby explaining our deft handling
of furniture and friends.
What does this have to do with real science? Apart, that is, from help-
ing to determine the optimal placement of heavy lab equipment on three-
legged tables? Plenty, many writers have thought. Polanyi argued that it
is by way of intuition that scientists assess the prima facie plausibility of
a hypothesis, and so decide to devote to it the time, energy, and money to
formulate, develop, and test it. Rohrlich (1996) adds that great discover-
ies are made by scientists who use their intuition to distinguish between
shortcomings in a theory that may be safely ignored and shortcomings
that shred a theory’s credibility. Finally, it is received wisdom among those
concerned with model-building in science that decisions as to what is es-
sential and what may be omitted from a model are frequently made using
a faculty much like physical, or biological, or social intuition.
This story is, as yet, less supported by evidence than by . . . intuition.
It is unclear, in particular, that there is a close relation between the abili-
ties used by ordinary people to navigate the everyday physical, biologi-
cal, and social world, and the abilities used by experienced scientists (let
alone great scientists) to assess the prima facie plausibility of hypothe-
ses, models, and theories.
Nevertheless, were Polanyi, Rohrlich, and the other intuitionists to be
correct, there would be a largely unexplored dimension along which sci-
ence and scientific progress might be understood. Theory construction, in
that case, even at the highest levels, need not be relegated to the shadow-
land of “insight” or “genius”: on the contrary, some ways in which mod-
els and theories are recruited and judged fit for duty will be as amenable
to study as any other psychological process. Further, to study such pro-
cesses, we need not hunt down distinguished scientists and imprison
them in mri machines, since the same patterns of thought are to be found
in college sophomores and other willing, plentiful subjects. All that is
needed is a bit of history-and-philosophy-of-science glue to hold every-
thing together.
This book is that glue. Or at least, it is the glue for an investigation into
one particular aspect of physical intuition: the ability of scientists and or-
dinary people to look at a physical scenario—or a biological scenario, or
a sociological scenario—and to “see” the physical probabilities of things
and, more generally, to “see” what properties a physical probability dis-

Physical Intuition 3
tribution over the outcomes of the scenario would have, without experi-
mentation or the gathering of statistical information. I call this ability to
infer physical probabilities from physical structure equidynamics.
With respect to equidynamics, everything promised above will be found
in the following chapters: an elucidation of the rules of thought by which
regular people, without the help of statistics, discern physical probabilities
in the world; an account of the way in which these everyday equidynamic
rules contribute to science by guiding judgments of plausibility and rele-
vance when building models and hypotheses, particularly in physics and
biology; and an account of the historical role played by equidynamic
thinking in great discoveries.
It is with a very great discovery based almost entirely on equidynamic
intuition that I begin . . .

7
Why does atmospheric pressure decrease, the higher you go? Why does
sodium chloride, but not silver chloride, dissolve in water? Why do com-
plex things break down, fall apart, decay? An important element of the
answer to each of these questions is provided by statistical mechanics,
a kind of physical thinking that puts a probability distribution over the
various possible states of the microscopic constituents of a system—over
the positions and velocities of its molecules, for example—and reasons
about the system’s dynamics by aggregating the microlevel probabilities
to determine how the system as a whole will most likely behave.
At the core of statistical mechanics are mathematical postulates that
specify probability distributions over the states of fundamental particles,
atoms, molecules, and other microlevel building blocks. The discovery of
statistical mechanics, this theoretical scaffold that now supports a great
part of all physical inquiry, began with the public unveiling of the first
attempt to state the exact form of one of these foundational probabilistic
postulates.
The date was September 21, 1859. At the meeting of the British Asso-
ciation for the Advancement of Science in Aberdeen, the Scottish physi-
cist James Clerk Maxwell read a paper proposing a probability distribu-
tion over the positions and velocities of the molecules of a confined gas at
equilibrium—that is, a distribution that the molecules assume in the
course of settling down to a steady statistical state (Maxwell 1860).1
According to the first part of Maxwell’s hypothesis, a molecule in a gas
at equilibrium is equally likely to be found anywhere in the space available.
The audience would have been neither surprised nor impressed by this sug-
gestion, as it echoed similar assumptions made by previous scientists
working on kinetic theory, and more important, conformed to everyday
1
T H E A P R I O R I S T

1 8 5 9
8
experience (it has never seemed to be harder to get a lungful of air at one
end of a room than at the other).2
What was dramatic about Maxwell’s
hypothesis was its second part, which went quite beyond the observed
properties of the atmosphere and other gases to propose that the compo-
nents of a gas molecule’s velocity are each described by a Gaussian proba-
bility distribution, from which it follows that the probability distribution
over the magnitude of velocity—the distribution of absolute molecular
speed v—has a probability density of the form av e bv
2 2
−
, where a and b are
constants determined by the mass of the gas molecules and the tempera-
ture of the gas. Figure 1.1 depicts the characteristic form of such a density.
The mathematician and historian of science Clifford Truesdell re-
marked that Maxwell’s derivation of the velocity distribution constitutes
“one of the most important passages in physics” (Truesdell 1975, 34). It
paid immediate dividends: Maxwell’s statistical model predicted that a
gas’s viscosity is independent of its density, a result that at first seemed
dubious to Maxwell but which he then confirmed experimentally. More
important, the model as later developed by Maxwell and Ludwig
Boltzmann yielded a quantitative explanation of entropy increase in gases,
along with the beginnings of an understanding of the microlevel founda-
tion of entropy. Finally, the ideas inherent in the model were applied, again
beginning with Maxwell and Boltzmann but also later by J. W. Gibbs and
others, to a wide range of physical systems, not limited to gases. Statisti-
cal mechanics in its mature form had by then emerged, a vindication of
Maxwell’s early conviction that “the true logic for this world is the cal-
culus of probabilities” (Garber et al. 1986, 9).
Maxwell’s proposed velocity distribution for gas molecules, the first
tentative step toward modern statistical physics, need not have been ac-
curate to have had the influence that it did. It would have been enough
to point the way to quantitative thinking about physical probabilities;
Figure 1.1: Maxwell’s proposed distribution over molecular speed

The Apriorist 9
further, some results about gases, including the viscosity result, do not
depend on the exact form of the distribution. Nevertheless, the velocity
distribution was quite correct.
Direct confirmation of the distribution’s correctness did not arrive
until long after Maxwell’s death when, in 1920, Otto Stern applied his
“method of molecular rays” to the problem. Stern’s apparatus allows a
brief “puff” to escape from a gas of heated metal atoms; the velocities of
the molecules in this burst are then measured by rapidly drawing a sheet
of glass or something similar past the site of the escape. When they hit
the sheet the molecules come to rest, forming a deposit. Because the fast-
est molecules in the burst reach the sheet first and the slowest molecules
last, and because the sheet is moving, the fast molecules will deposit
themselves at the head of the sheet, the slow molecules at its tail, and
molecules of intermediate speeds in between at positions proportional
to their speed. The distribution of molecular velocity will therefore be
laid out along the sheet in the form of a metallic film. Stern showed that
the density of the film mirrored the density of Maxwell’s velocity distri-
bution (Stern 1920a, 1920b). He considered this as much a test of his
new apparatus as a test of Maxwell’s hypothesis, so secure was the theo-
retical status of the Maxwell distribution by this time.
If Maxwell did not measure the velocities of molecules himself, how
did he arrive at his probabilistic postulate? Did he work backward from
the known behavior of gases, finding the only distribution that would
predict some known quirk of gaseous phenomena? Not at all. Maxwell’s
derivation of the velocity distribution rather appears to be entirely a pri-
ori. “One of the most important passages in physics”—the seed that
spawned all of statistical mechanics—was a product of the intellect alone,
empirically untempered. Could this semblance of aprioricity be the real
thing? Could Maxwell have directly intuited, in some quasi-Kantian
fashion, the precise shape of the actual distribution of molecular veloc-
ity? Did he make a lucky guess? Or was something else going on? This
book aims to answer these questions and to generalize the answer to
similar probabilistic inferences in other domains of scientific inquiry.
Maxwell’s derivation of the velocity distribution is, on the surface, short
and simple. Grant that gases are made up of myriad particles, microscopic
and fast-moving. (This assumption, still controversial in 1859, was of

1 8 5 9
10
course empirical; what was a priori about Maxwell’s reasoning, if any-
thing, was the statistical component of his model of a gas.) What can be
supposed about the velocities of these molecules at equilibrium?
Three things, Maxwell suggests. First, the probability of a molecule
having a certain velocity will be a function of the magnitude of that
velocity alone; or, in other words, molecules with a given speed are
equally likely to be traveling in any direction. Second, the probability
distributions over each of the three Cartesian components of a mole-
cule’s velocity—its velocities in the x, y, and z directions—are the same.
Third, the three Cartesian components of velocity are independent of
one another. Learning a molecule’s velocity in the x direction will, for
example, tell you nothing about its velocity in the y direction.
These assumptions put what might look like rather weak constraints
on the molecular velocity distribution. In fact, they are anything but
weak: there is only one distribution that satisfies them all. Maxwell con-
cludes that the distribution of molecular velocity must assume this form.
(The mathematical details of the derivation are set out in the appendix
at the end of this chapter.)
Naturally, then, the reader asks: what is the basis of the three assump-
tions, so strong as to uniquely determine the velocity distribution, and
correctly so? Maxwell makes the first assumption, that the probability of
a velocity depends only on its magnitude, on the grounds that “the direc-
tions of the coordinates are perfectly arbitrary” (Maxwell 1860, 381),
that is, that for the purpose of providing a Cartesian representation of
molecular velocity, you could choose any three mutually orthogonal lines
to be the x, y, and z axes. He does not provide an explicit justification for
the second assumption, but he has no need of one, because it follows
from the other two (Truesdell 1975, 37). The rationale for the third as-
sumption, the independence of a molecular velocity’s three Cartesian
components, is as follows: “The existence of the velocity x does not in any
way affect that of the velocities y or z, since these are all at right angles to
each other and independent” (p. 380).
The premises from which Maxwell deduces his velocity distribution
are two, then: first the conventionality of, and second the mathematical
independence of, the three axes that contribute to the Cartesian represen-
tation of velocity. Each of these is, if anything is, a conceptual truth. Max-
well thus appears to have derived his knowledge of the actual distribution
of molecular velocities on entirely a priori grounds: apart from the empiri-
cal presuppositions of the question—that there are molecules and that

The Apriorist 11
they have a range of velocities—the answer seems to rest on abstract philo-
sophical or mathematical truths alone, floating free of the particular, con-
tingent physics of our world and readily apparent to the armchair-bound
intellect.
Had an eminent mind claimed physical knowledge based on pure re-
flection in the seventeenth century, you would not be surprised; nor would
you expect their claims, however historically and intellectually interesting
they might be, to correspond to physical reality. But it was the nineteenth
century. And Maxwell was right.
This ostensible vindication of untrammeled apriorism demands a closer
look.What was going on under the surface of Maxwell’s text? How might
considerations other than purely conceptual observations about the Car-
tesian system of representation have guided Maxwell’s argumentation, or
given him the confidence to present the conclusions that he did?
There are three places to search for clues, to which the next three chap-
ters correspond. First there are what I will call, in the broadest sense, the
historical facts, by which I mean primarily the scientific writings of Max-
well and his predecessors. Second, there are what I will call the logical
facts, by which I mean the facts as to what forms of reasoning, a priori
or otherwise, give reliable or at least warranted information about phys-
ical probability distributions. Third, there are the psychological facts,
the facts about what forms of reasoning to conclusions about probabil-
ity distributions—if any—are typically found in human thought.
Appendix: Maxwell’s Derivation of the Velocity Distribution
Maxwell supposes that the same density f(⋅) represents the probability
distribution over each of the three Cartesian components of a molecule’s
velocity. Further, the components are stochastically independent, so the
probability density over velocity as a whole is a simple function of the
densities over the components:
F(x,y,z) = f(x)f(y)f(z)
At the same time, Maxwell assumes that the probability distribution
over velocity as a whole depends only on velocity magnitude, and so
may be represented as a function of magnitude squared alone:
F(x,y,z) = G(x2
+ y2
+ z2
)
for some choice of G. The form of the distribution, then, is such that

1 8 5 9
12
G(x2
+ y2
+ z2
) = f(x)f(y)f(z)
for some choice of f and G.
Surprisingly, only one choice of f satisfies this constraint:
f x aebx
( )
2
=
As with any probability density, the area under this function must sum to
one, from which it follows that the coefficient of the exponent is negative.
The component density f is therefore a Gaussian distribution.

13
Why did Maxwell find his derivation of the velocity distribution con-
vincing or, at least, plausible enough to present for public consumption?
Why did a significant portion of his public—the physicists of his day—
regard it in turn as a promising basis for work on the behavior of gases?1
And how did he get it right—what was it about his train of thought, if
anything, that explains how he hit upon a distribution that not only com-
manded respect from his peers but accurately reflected reality? The same
distribution made manifest in Otto Stern’s molecular deposits sixty years
later?
Under the heading of history, I look for answers to these questions not
only in the broad currents of nineteenth-century scientific thinking, but
also in Maxwell’s own writings, and in particular—and, as it will turn
out, most importantly—in the other parts of the paper in which he de-
rives his velocity distribution.
2.1 Proposition IV
Begin with “history” in the narrowest sense, that is, with Maxwell’s overt
reasons for postulating the velocity distribution. The derivation occupies
just the few paragraphs that make up proposition iv of Maxwell’s 1859
paper on gases (Maxwell 1860). Put on hermeneutic blinders and imag-
ine for a moment that nothing else exists, that proposition iv contains
everything relevant to Maxwell’s reasoning. How convincing, and how
truth-conducive, is the derivation?
It is remarkably weak.
The first premise of the derivation, that the probability of a velocity
depends only on its magnitude, is equivalent to the assumption that the
2
T H E H I S T O R I C A L WAY

1 8 5 9
14
distribution over velocity is spherically symmetrical: it looks the same
from all directions, or in other words, however you rotate it.
The assumption has a certain plausibility. But Maxwell’s reason for
advancing it is unconvincing. He observes that the directions of the Car-
tesian axes used to represent velocity are arbitrary, something that, being
a conceptual truth about the nature of Cartesian representation, is true
for any probability distribution over velocity, or indeed, over any physi-
cal quantity represented in three-dimensional space. But it is obvious
that not all such distributions will have spherical symmetry. The distri-
bution over the positions of gas molecules in a significant gravitational
field, for example, tails off with height (that is, distance from the gravi-
tational attractor).2
Thinking about this and other cases, you should see that the symmetry
assumption is entirely unmotivated by the arbitrariness of the coordinate
system. Something else must give the assumption its credibility, presum-
ably something that is present in a normal enclosed gas but not in a gas
subject to a gravitational field or similar force—something physical, then.
So facts about physical symmetries or asymmetries play a role in per-
suading us to accept Maxwell’s first premise. What facts? I can hardly
wait to answer this question, but for now it is postponed: this is the his-
tory chapter, in which I restrict myself to justifications offered on the
record, or at least hinted at, by Maxwell and his contemporaries.
The second Maxwellian premise, the stochastic independence of the
Cartesian components of velocity, has less prima facie plausibility than
the first. Maxwell’s justification of the premise does not improve the
epistemic situation: he offers nothing more than the observation that the
Cartesian components are independent in the mathematical sense (a con-
sequence of the orthogonality of the axes). This is better than wordplay,
since mathematical independence is (more or less) a necessary condition
for stochastic independence, but it is not much better, being far from a
sufficient condition. Maxwell himself later conceded that his reasoning
was at this point “precarious” (Maxwell 1867, 43).
Yet even Maxwell’s own later misgivings cannot annul the fact that his
reasoning, however logically insecure, led him to true conclusions with
momentous consequences—momentous in part because a few of his
more extraordinary readers, not least Boltzmann, thought that he had
gotten something important right.
A close examination of Maxwell’s official reasons for making the as-
sumptions he does, then, aggravates rather than alleviates the puzzle.

The Historical Way 15
Maxwell not only made a great empirical discovery by way of an en-
tirely a priori argument—it was a bad a priori argument.
Again, it should be asked: did he just get lucky? Did he, in the course
of playing with statistical models constructed more with the aim of sim-
plicity than veracity, stumble on the truth? But then how did he recog-
nize it for the truth, or at least, for a serious possibility? How did his
peers do the same?
2.2 The Zeitgeist
Maxwell’s derivation was not original—not in its mathematical skele-
ton. Partway through a long 1850 review of a book by Adolphe Quetelet,
John Herschel used a similar argument to justify the supposition that
errors in a wide variety of scientific measurements assume a Gaussian
distribution, so vindicating the use of the “method of least squares” to
deal with measurement error (Herschel 1850; see also section 11.1 in
this volume).
Herschel reasons as follows. He compares the process of making astro-
nomical measurements—of pointing a telescope at a celestial object to
measure, say, its position from day to day—with a marksman firing a ri-
fle at a target or a scientist repeatedly dropping a ball from a great height
attempting to hit a specified point on the ground below. Each of these
processes will be subject to error, and in each case three assumptions can,
according to Herschel, be made about the error:
1. There is a probability distribution over the possible errors that takes
the same mathematical form for every such process; that is, there is
a single probability distribution describing the errors in rifle-firing,
telescope-pointing, and ball-dropping (though presumably the pa-
rameters of the distribution vary from case to case).
2. The probability of a given error depends only on its magnitude; thus,
an error of a given magnitude is equally likely to be in any direction, or
equivalently, the distribution over errors in the two-dimensional tar-
get space has circular symmetry.
3. The Cartesian components of any given error are independent.
From the second and third premises, mathematically equivalent to the
premises of Maxwell’s derivation (though Herschel’s measurement errors
are represented in two-dimensional space, whereas molecular velocities

1 8 5 9
16
are represented in three dimensions), Herschel concludes that the distribu-
tions over the Cartesian x and y components of error are Gaussian, and
consequently that the distribution over the magnitude m of error has the
form ame bm2
−
.
The principal difference between Herschel’s and Maxwell’s derivations,
aside from the subject matter, is in the grounds that they give for their as-
sumptions. Maxwell’s grounds are conceptual truths about Cartesian
representation. Herschel adds to the mix considerations of ignorance.
First, he argues that the probability distribution over errors will be the
same regardless of the source of the error, because we are equally ignorant
of the causes in all cases; for example, because we know nothing to distin-
guish the causes of error in marksmanship and telescopy (or at least, we
know nothing distinctive about the statistical distribution of the causes),
we should suppose that the same probability distribution describes both.
Second, he gives the same reason—“our state of complete ignorance of
the causes of error, and their mode of action” (p. 20)—for making the as-
sumption of circular symmetry, that is, for supposing that the probabil-
ity distribution depends only on the magnitude of the error. (Precisely
why our ignorance justifies the assumption that the probability of an
error may depend on its magnitude but not on its direction, rather than,
say, vice versa, is unclear, though of course there are simple nonepistemic
reasons why the probability must eventually taper off with increasing
magnitude.) Ignorance does not come into Herschel’s justification of the
component independence assumption, which appears to be similar to
the justification offered by Maxwell.3
Despite the central role played by ignorance in grounding his Gauss-
ian error distribution, Herschel expects actual frequencies of errors to
conform to the distribution:
Hence this remarkable conclusion, viz. that if an exceedingly large number of
measures, weights, or other numerical determinations of any constant magni-
tude, be taken,—supposing no bias, or any cause of error acting preferably in
one direction, to exist— . . . the results will be found to group themselves . . .
according to one invariable law of numbers [i.e., the law of errors] (Herschel
1850, 20–1).4
For this attempt to create statistical knowledge from an epistemic void,
Herschel was roundly criticized by R. L. Ellis (1850), who made many

of the same objections you would hear from a philosopher of science
today (see section 3.2).
Was Maxwell influenced by Herschel? Historians of physics agree that
he was likely familiar with Herschel’s argument by the time he derived
the velocity distribution (Garber 1972). He may well have taken the idea
from Herschel, then, while purging it of its subjectivist elements, that is,
of its reliance on considerations of ignorance. It is possible also that Her-
schel’s immense standing in the world of British science gave Maxwell
some confidence in the force of the argument. (It is unclear whether he
was familiar with Ellis’s and others’ objections to Herschel.)5
However, it would surely have been reckless for Maxwell to publish
his theory of gases on these grounds alone. His decision to delete the sub-
jectivist aspect of Herschel’s argument suggests that he did not find it en-
tirely satisfactory as it stood. More important, whereas Herschel was ex-
plainingastatisticalpatternforwhichempiricalevidencewasaccumulating
rapidly—the Gaussian distribution of measurement error—Maxwell was
predicting a statistical pattern for which there was no empirical evidence
whatsoever.
Further, that a star-struck Maxwell swallowed Herschel’s argument
without complaint hardly explains the extraordinary fact that his subse-
quent reasoning led him directly to the velocity distribution’s true form. It
would be a mistake to assume that Maxwell’s success could only be ex-
plained by his using an infallible or entirely rational method, but it would
be equally a mistake to attribute his success to pure chance, holding that it
was simply Maxwell’s good fortune that fashionable ideas about the dis-
tribution of errors happened to point to the right probability distribution
for gaseous molecular velocities. Or rather, it would be a mistake to settle
for the luck hypothesis without first looking harder for alternatives.
The sociocultural sources of Maxwell’s 1859 derivation surely extend
beyond Herschel’s review, but how far and in what directions we may
never know; certainly, historians have uncovered little else about the pos-
sible influences on the argument’s specific form. Let me therefore return to
textual analysis of Maxwell’s 1859 derivation of the velocity distribution.
2.3 The Road to Proposition IV
Maxwell’s apparently a priori discovery of the velocity distribution ap-
pears in the 1859 paper’s proposition iv. If this passage deduces from

1 8 5 9
18
first principles the conceptual foundation for everything that is to fol-
low, what is in propositions i through iii?
Proposition iv derives the form of the molecular velocity distribution
for a gas that has settled down to a statistically steady state. Propositions
i, ii, and iii purport to establish the existence of such a steady state, that
is, they purport to establish that over time, the distribution of velocities
in a gas will converge to a unique fixed distribution—that the velocity
distribution has a global, stable, equilibrium. The first three propositions
play a similar role, then, to Herschel’s argument that, because of our ig-
norance of the causes of error, there is a unique distribution over mea-
surement errors of all types.
Maxwell’s argument for the equilibrium runs as follows. Proposition i
lays out the physics of collisions between perfectly elastic hard spheres—
the physics of idealized billiard balls. (Maxwell has already remarked
in the paper’s introduction that his conclusions will apply equally to
particles that do not collide but that repel one another by way of strong
short-range forces.) As yet, no statistical considerations are introduced.
Proposition ii aims to calculate the probability distribution over the
rebound angle resulting from such a collision. Fix the frame of reference
so that one sphere is not moving. Then the other sphere, if there is to be
a collision, must be moving toward the fixed sphere. More specifically,
the moving sphere’s center must pass through a circle orthogonal to its
direction of motion, a circle whose center lies on a line parallel to the
direction of motion emanating from the center of the fixed sphere, and
whose radius is equal to the sum of the two spheres’ radii, as shown in
figure 2.1. Call this the approach circle. (The notional approach circle
may lie at any point between the two spheres; it does not matter where.)
Maxwell now introduces a statistical postulate: he assumes that, condi-
tional on the occurrence of a collision, the moving sphere’s center is
equally likely to have passed through any point of the approach circle.
Call the point where the sphere’s center passes through the circle the
sphere’s approach point; then Maxwell’s assumption is that any approach
point within the circle is equally probable, or in other words, that the
probability distribution over the approach point, conditional on a colli-
sion’s taking place, is uniform. (Like Maxwell, I will not stop to ask in
virtue of what facts there could be such a probability distribution in a
world whose laws were, as Maxwell believed, deterministic.) From the
uniformity assumption and the physics of collisions, Maxwell shows that

the moving sphere is equally likely to rebound in any direction, that is,
that its velocity after the collision is equally likely to be in any direction.6
Proposition iii brings together the deterministic physics of collisions
with the rebound angle equiprobability derived in proposition ii; it states
that the velocity of a sphere after a collision with another sphere is the
sum of the velocity of the spheres’ center of mass and a velocity deter-
mined by the impact itself, which will be with equal probability in any
direction, relative to the center of mass (since the rebound angle in the
center-of-mass frame of reference will be the same as in proposition ii’s
fixed-sphere frame). In effect, the collision subjects the velocity of the
sphere to a random adjustment.
Maxwell then concludes:
If a great many equal spherical particles were in motion in a perfectly elas-
tic vessel, collisions would take place among the particles, and their velocities
would be altered at every collision; so that after a certain time the vis viva
[kinetic energy] will be divided among the particles according to some regu-
lar law, the average number of particles whose velocity lies between certain
limits being ascertainable, though the velocity of each particle changes at
every collision (Maxwell 1860, 380).
In other words: the cumulative effect of the random adjustments of ve-
locity due to manifold collisions will converge on a single distribution
(the “regular law”). Maxwell’s proposition iv follows, opening with the
stated aim of finding “the [velocity distribution] after a great number of
Figure 2.1: The approach circle: the two spheres collide just in case the center of
the moving sphere (right) passes through the approach circle

1 8 5 9
20
collisions among a great number of equal particles”(where“equal”means
“equally massive”). At this point, Maxwell presents the apparently a pri-
ori derivation of the velocity distribution.
Why did Maxwell think that the aggregate effect of many random ve-
locity adjustments would impose a fixed and lasting distribution over
velocity? He does not say, but the conclusion hardly follows from its
grammatical antecedent, that the molecules’ velocities “would be altered
at every collision.”
You might guess that Maxwell is reasoning thus. The velocity of a par-
ticle, after a large number of collisions, will be the sum of its initial veloc-
ity and the changes in velocity effected by the collisions. After many colli-
sions, the initial velocity will comprise a negligible portion of this sum—it
will have been “washed out.” Thus a molecule’s velocity, after a suffi-
ciently long period of time, will be for all practical purposes determined
by the distribution of the changes in velocity effected by collisions. If there
is a fixed distribution over these changes that does not itself depend on the
molecules’ initial conditions, then there will be, “after a certain time,” a
fixed distribution over the velocities themselves.
In supposing the existence of the latter fixed distribution—the distribu-
tion over changes in velocity—Maxwell might have taken some comfort
from de Moivre’s theorem and Laplace’s more general central limit theo-
rem showing that iterated random fluctuations of the right sort converge,
if independent, to a Gaussian distribution. (To secure independence, he
would have to assume that the approach points for any two collisions are
independent of each other, a postulate that seems as plausible as the uni-
formity assumption itself.) But if Maxwell saw his conclusion as hinging
on these powerful mathematical results, it is curious that he does not
think to say so.
I will argue eventually (see section 8.4) that Maxwell’s reasoning does
turn on “washing out” of the sort described above, and I will explain
why many major premises of the washing-out argument are nevertheless
missing from the text.
Until then, let me rest with a provisional conclusion. Propositions i
through iii have an important role to play in Maxwell’s derivation of the
velocity distribution; their function is to establish that there is a unique
equilibrium distribution to derive. There is clear textual evidence for this
claim: as noted above, after the final sentence of proposition iii declares
that the velocity distribution of “a great many . . . particles” settles down
as a result of collisions to an equilibrium “after a certain time,” the first

sentence of proposition iv declares explicitly the intent to derive the
form of the velocity distribution resulting from “a great number of colli-
sions among a great number of . . . particles.” Maxwell’s later summary of
the first part of the paper confirms in passing the role of the approach to
equilibrium (1860, 392).7
Though the argument for the existence of the
equilibrium distribution is perhaps somewhat obscure, there can be no
question that Maxwell attempted such an argument and that he regarded
it as an integral part of the derivation of the velocity distribution.
If my provisional conclusion is correct, then Maxwell’s derivation of
the distribution is not, after all, a priori, for one and a half reasons. First,
the derivation of rebound-angle equiprobability requires assumptions
about the physics of intermolecular collisions. Although Maxwell’s as-
sumptions were based on surmise rather than empirical testing, he of
course made no claim to discern the physics a priori. Second—and this
is the half-reason—the derivation requires assumptions about the prob-
ability distribution over approach points. Unlike the probabilistic as-
sumptions in proposition iv, Maxwell does not base these on conceptual
truths about Cartesian representation. Nor does he give them any other
rationale. He simply asserts in proposition ii that “within [the approach]
circle every position is equally probable,” as if it were indisputable. In-
disputable because a priori? Maxwell does not say.
I now want to go back to reexamine Maxwell’s grounds for making the
statistical assumptions that serve as the premises of his Herschelian argu-
ment in proposition iv, namely, the equiprobability of velocities of equal
magnitude and the independence of the three Cartesian components of
velocity. The probabilistic reasoning of propositions i through iii, I will
propose, provides an alternative and more plausible grounding for the
proposition iv assumptions than Maxwell’s official grounding in the for-
mal properties of Cartesian representation. Although Maxwell did not
make this alternative grounding explicit, it played a role, I conjecture, in
convincing him and some of his readers of the validity of the proposition
iv assumptions, and equally importantly, it helps to explain how Max-
well so easily hit upon the truth.
Begin with the assumption that a gas molecule with a given speed is
equally likely to be traveling in any direction—that is, the assumption that
the velocity distribution has spherical symmetry. There is a justification

1 8 5 9
22
for this assumption that closely parallels Maxwell’s putative rationale, sug-
gested above, for the existence of a fixed long-run velocity distribution. It
goes as follows. From propositions ii and iii, the velocity distribution over
changes in a molecule’s velocity is spherically symmetric. It follows, if
these changes are stochastically independent, that the distribution over a
series of such changes is also spherically symmetric. Since the velocity dis-
tribution is determined, in the long run, by this spherically symmetric
distribution—initial conditions are “washed out”—it too must be spheri-
cally symmetric.
The argument is not quite correct, however: the distribution over
changes in velocity relative to a colliding pair’s center of mass is spherical,
but different collisions have different centers of mass. Before aggregating
the changes, then, they ought to be converted into a common frame of
reference; the obvious choice would be the rest frame with respect to
which velocities are represented by the velocity distribution. This is not
straightforward,as subsequent attempts to provide more rigorous grounds
for kinetic theory have shown. The task is not impossible, but it is proba-
bly not reasonable to suppose that it was accomplished by Maxwell.
Another, surer line of thought was available to Maxwell, however. He
had concluded, at the end of proposition iii, that the distribution of mo-
lecular velocities converges over time to a unique distribution—a global,
stable, equilibrium. Once established, this distribution, being a stable equi-
librium, remains the same; in particular, it does not have any tendency to
change in the short term, minor fluctuations aside. Therefore, a distribu-
tion of velocities (or positions) that does change in the short term cannot
be the equilibrium distribution, whereas a distribution that does not change
must be the equilibrium distribution, since there is only one equilibrium.
By examining short-term trends in the behavior of different distributions,
then, it is possible to learn the properties of the equilibrium distribution.
Suppose that a gas’s velocities are distributed so that the first Her-
schelian premise is false: among molecules traveling equally fast, some
directions of travel are more probable than others. Because of rebound
angle equiprobability, intermolecular collisions will very likely soon begin
to undo this bias: some more probable directions of travel will become
less probable than before, while some less probable directions become
more probable than before. No biased distribution is stable; therefore, the
equilibrium distribution cannot be a biased distribution. It must be a dis-
tribution in which a molecule traveling at a given speed is equally likely to
be traveling in any direction.8

Or you might reason the other way. A distribution in which velocities
of equal magnitude are equiprobable is stable, because no particular di-
rectional bias is favored in the short term by the stochastic dynamics of
collisions established in propositions i to iii. Therefore, such a distribu-
tion must be the equilibrium distribution.
Note that even the conjunction of the above two suppositions about
short-term stability, amounting to the proposition that the only stable
velocity distribution is one in which direction given magnitude is equi-
probable, is insufficient to establish the existence of a global equilib-
rium: it is consistent with the possibility that for many initial states, a
gas’s velocity might never equilibrate, that is, might never settle down to
a single, fixed distribution.
The same kind of reasoning may be used to justify the second Her-
schelian premise, that the Cartesian components of velocity are stochas-
tically independent. In this case, the relevant short-term probabilistic
trend breaks down correlations among the Cartesian components of a
molecule’s velocity. As a result of this trend, no distribution with corre-
lated components has short-term stability; therefore, no such distribu-
tion is the equilibrium distribution, and so the components in the equi-
librium distribution are independent.The trend to Cartesian dissociation
is, I think, less evident than the trend dissociating direction and magni-
tude of speed; further, Maxwell’s investigations in propositions i to iii
do less to establish its existence than they do to establish the existence of
direction/magnitude dissociation. But still, such a trend can be faintly
discerned by the unaided human intellect.
What evidence is there that Maxwell’s confidence in the Herschelian
premises was based on this line of reasoning, this relationship between
long-term equilibrium and short-term dynamics? In proposition iv, no
evidence whatsoever. But later in the 1859 paper, similar conclusions are
based on argumentation of precisely this form.
In proposition xxiii, for example, Maxwell sets out to show that the
kinetic energy of a gas composed of nonspherical molecules will become
equally distributed among the three Cartesian components of (transla-
tional) velocity and the three Cartesian components of angular velocity
“after many collisions among many bodies.” Invoking the velocity distri-
bution as a premise, he demonstrates that only an equal distribution of
energy between the translational and angular velocities is stable in the
short term, and concludes that such a distribution will be “the final state”
of any such gas (Maxwell 1860, 408–409). Maxwell therefore reasons,

1 8 5 9
24
in this passage, from facts about short-term stability and instability to the
properties of the equilibrium distribution, and does so by way of an argu-
ment that presumes the existence of a unique, global, stable equilibrium—
his “final state.” Without this assumption the argument fails, as in itself it
gives no reason to suppose that the state of equal energy distribution is
stable, let alone that a system starting out in an arbitrary nonequilibrium
state will eventually find its way to the equilibrium.
With a little more hermeneutic effort, which I leave as an exercise to
the reader, the same kind of reasoning can be discerned in proposition
vi; what is more, Maxwell later used a similar approach to provide a
new foundation for the velocity distribution in his “second kinetic the-
ory” (Maxwell 1867).
It would be peculiar if Maxwell, having developed a stochastic dynam-
ics for molecular collisions and having used it to argue for the equilibra-
tion of the velocity distribution in propositions i to iii, and then having
used both the short-term dynamics and the putative fact of global equili-
bration to derive properties of the equilibrium distribution in other parts
of his paper (propositions vi and xxiii), entirely ignored the relevance of
the same kind of argument for his Herschelian posits in proposition iv.
Although the official justification of the posits makes no mention of
such considerations, then, there lie in Maxwell’s text abundant logical,
physical, and probabilistic materials for the construction of an alterna-
tive, unofficial justification for the posits that is considerably more con-
vincing. It is no great leap to suppose that the force of this unofficial
argument, however dimly perceived, played some role in encouraging
Maxwell to continue his researches and to publish his paper, and also
perhaps in encouraging his more perceptive readers to take Maxwell’s
statistical model seriously.
How did Maxwell’s paper come to contain two rival arguments? I
suspect that the paper was shaped originally around what I am calling
the unofficial argument, that is, the reasoning based on the equilibrating
effect of the accumulation of many independent adjustments of molecu-
lar motion described in proposition iii. Maxwell saw that the argument
was not mathematically complete and, dissatisfied, at some point in-
serted the alternative a priori argument for the Herschelian posits, sever-
ing proposition iv’s connection to the preceding three propositions and
introducing an official argument whose mathematics is as ineluctable as
its logic is inscrutable. If this is correct then the official, a priori argu-
ment played no role whatsoever in Maxwell’s discovery of the velocity

distribution; it is merely a post hoc philosophical papering over of the
flawed but vastly more fruitful unofficial argument that guided Maxwell
to the insights on which statistical physics was built.
Putting these speculations aside, it is in any case the unofficial argu-
ment, I have suggested, that explains both the rhetorical power and the
truth-conduciveness of Maxwell’s reasoning. But this is possible only if
the lacunae in the argument for the existence of a global equilibrium were
somehow logically and psychologically patched and only if the probabi-
listic posits upon which the equilibration argument is itself based, the
uniform distribution of and the stochastic independence of the pre-
collision approach points, were themselves plausible and close enough to
true.
You might think that the uniformity and independence posits alone
sink the story. What reason is there to suppose that they are correct, let
alone transparently so to nineteenth-century readers? We have gone from
probabilistic postulates with a questionable basis—the postulates of prop-
osition iv with their foundation in conceptual truths about Cartesian
representation—to probabilistic postulates with no visible basis at all. Is
this an improvement?
History can only get you so far. So much reasoning goes unrecorded—
neither permanently recorded in the annals of science, nor even ephem-
erally recorded in the mind’s eye. So much reasoning is simply uncon-
scious. Other methods are needed to dig it out.

26
The scientific power of Maxwell’s first paper on statistical physics sprang
in large part, I have proposed, from an unofficial, alternative argument in
which the mysterious premises of proposition iv, the equiprobability of
velocity’s direction given its magnitude and the independence of its Car-
tesian components, function as intermediate steps rather than as founda-
tions. The real foundations are specified in propositions i through iii.
They are of two kinds: dynamic facts about the physics of collisions, and
further probabilistic posits.
The more salient of the new probabilistic posits is the assumption that
a molecule is in some sense equally likely, en route to a collision, to pass
through any point on the approach circle (section 2.3). This is presum-
ably a special case of a more general assumption: the probability distri-
bution over molecular position is approximately uniform over any very
small area.
I have also tentatively attributed to Maxwell a second probabilistic
posit, concerning the stochastic independence of the approach points of
any two collisions. Maxwell does not make this assumption explicit, but
it or something similar must be attributed to him to make sense of his
reasoning about the cumulative effects of many collisions. I will not
make any further specific claims about the form of Maxwell’s indepen-
dence posit until chapters 7 and 8, but bear in mind its existence.
Call these two posits together the assumption of the microequiprob-
ability of position. (Microequiprobability incorporates some sort of
stochastic independence, then, even though independence makes no
contribution to its name. Characterizations of new technical terms in-
troduced in this book may be found in the glossary.) What advantage is
3
T H E L O G I C A L WAY

The Logical Way 27
gained by basing the derivation of the velocity distribution on the mi-
croequiprobability of position rather than the proposition iv posits?
Microequiprobability has a kind of intuitive rightness that the proposi-
tion iv posits lack. Simply to assume the independence of the Cartesian
components of velocity is rebarbative; to assume it on the grounds that
the components are mathematically independent is if anything even more
so. (Better, sometimes, not even to try to explain yourself.) To assume that
positions are microequiprobable seems, by contrast, rather reasonable.
Why? What is the source of our epistemic comfort? Is it emotional,
rhetorical, mathematical, physical? Or philosophical? Logical?
3.1 Doors and Dice
You must choose between three doors. Behind one is a hungry saber-
toothed tiger. Behind another is a lump of coal. Behind the third is a
well-upholstered endowed chair. That is all you know. Is there any rea-
son to prefer one door over the others? Apparently not. You should
consider the tiger equally likely to lurk behind any of the three doors;
likewise the chair.
You roll an unfamiliar looking die, in the shape of a dodecahedron
(figure 3.1). Its faces are numbered 1 through 12. In a series of rolls, is
Figure 3.1: A dodecahedral die

1 8 5 9
28
there any reason to expect one face—say, the 5—to turn up more often
than the others? Apparently not. You should consider a roll of 5 to be
just as likely as any other roll.
In both cases, you note some symmetry in your situation—the three
identical doors, the twelve identical faces—and from that symmetry you
derive a probability distribution that reflects the symmetry, assigning
equal probabilities to relevantly similar outcomes. This is the probability
distribution that seems “intuitive” or “reasonable” or “right” given the
symmetry.
Cognitive moves such as these—in which the mind goes from observing
a symmetry or other structural feature to imposing a probability
distribution—are traditionally thought to be justified, if at all, by some-
thing that philosophers now call the principle of indifference. The princi-
ple is a rule of right reasoning, an epistemic norm; its treatment thus falls
into the domain of probabilistic epistemology, broadly construed. It is
with logical or philosophical methods, then, that I will examine the pos-
sibility that the force of Maxwell’s microequiprobability posit, and so the
power and truth-conduciveness of his argument, rests on indifference.
The difficulties in interpreting the principle have prompted many epis-
temologists to give up on indifference altogether. That, I hope to per-
suade you, is a bad mistake. There are important forms of thought in
both the sciences and in everyday life that turn on reasoning from sym-
metries and other structural properties to probabilities; if we do not
understand such reasoning, we do not understand our own thinking.
That said, the principle of indifference is a chimera—a fantasy, but a
fantasy made up from real parts. For me, the more important of these
parts is a rule warranting the inference of physical probability distribu-
tions from physical structure; this, I will argue, is what gives Maxwell’s
microequiprobability posit its foundation. But the other part deserves
attention too; I will have something good to say about it in passing.
3.2 Classical Probability and Indifference
The historical principle of indifference, then known as the principle of
insufficient reason, was born twinned with the classical notion of prob-
ability, canonically defined by Pierre Simon Laplace as follows:
The theory of chance consists in reducing all the events of the same kind to
a certain number of cases equally possible, that is to say, to such as we may

The Logical Way 29
be equally undecided about in regard to their existence, and in determining
the number of cases favorable to the event whose probability is sought. The
ratio of this number to that of all the cases possible is the measure of this
probability, which is thus simply a fraction whose numerator is the number
of favorable cases and whose denominator is the number of all the cases
possible (Laplace 1902, 6–7).
The definition of classical probability and the historical principle of in-
difference are one and the same, then; consequently, if you apply the
principle correctly, you will have certain knowledge of the relevant clas-
sical probability distribution. Assuming, for example, that it is “equally
possible” that each of the dodecahedral die’s faces ends a toss upper-
most, you may apply the principle, or what amounts to the same thing,
apply the definition of classical probability, to gain the knowledge that
the classical probability of obtaining a 5 is 1/12.
The classical definition, and so the indifference principle, raise a num-
ber of difficult philosophical questions. First, Laplace’s two descriptions
of the conditions under which the definition can be applied—to “equally
possible” cases, and to cases “such as we may be equally undecided
about”—do not seem to be equivalent. The satisfaction of the latter con-
dition is a matter of mere ignorance, while the former appears to hinge
on a feature of the world and not of our knowledge state: immediately
before the quoted passage, Laplace talks about our “seeing that” several
cases are equally possible, and later he writes that, in the case where out-
comes are not equipossible, it is necessary to “determine . . . their respec-
tive possibilities, whose exact appreciation is one of the most delicate
points of the theory of chances” (p. 11). As many writers have observed,
classical probability itself inherits this apparent duality in its defining
principle (Hacking 1975; Daston 1988).
Second, it is clear that some differences between cases are irrelevant to
the application of the principle. That the faces of the dodecahedral die
are inscribed with different numbers ought not to affect our treating them
as “equally possible” or our being “equally undecided” about them. Why
not? What properties are and what properties are not relevant to deter-
mining the symmetries and other structural properties that fix the distri-
bution of probability?
Third, Laplace’s definition apparently makes it possible to derive
knowledge of the classical probability of an event on the grounds of
personal ignorance concerning the event. The less you know about a set

1 8 5 9
30
of events, it appears, the more you know about their probability. This
peculiarity seems positively objectionable if classical probabilities are
allowed to play a role in prediction or explanation. So Ellis, for exam-
ple, objects to Herschel’s basing a probability distribution over measure-
ment error on our ignorance of the causes of error, but then using that
distribution to predict—successfully!—the actual distribution of errors,
that is, the frequencies with which errors of different magnitudes occur
(section 2.2). Ex nihilo nihil, as Ellis (1850, 325) quite reasonably writes.
The same objection can be made to certain of Laplace’s own uses of the
principle.
These problems, though well known, have been eclipsed by a fourth,
the late nineteenth-century demonstrations that the indifference princi-
ple fails to deliver consistent judgments about the probability distribu-
tion over real-valued quantities, a result that has come to be known as
Bertrand’s paradox (von Kries 1886; Bertrand 1889).1
A simple version
of the paradox—a variation on Keynes’s variation on von Kries—is due
to van Fraassen (1989, 303–304). Consider a factory that produces only
cubes, varying in edge length from 1 centimeter (cm) to 3cm. What is the
probability of the factory’s next cube having edge length less than 2cm? It
seems that the indifference principle will advise me to put a uniform dis-
tribution over edge length, arriving at a probability of 1/2. But in my state
of utter ignorance, I might surely equally well put a uniform distribution
over cube volume. A cube with edge length less than 2cm will have a vol-
ume between 1 and 8 cubic cm, while a cube with edge length greater than
2cm will have a volume between 8 and 27 cubic cm. A uniform distribu-
tion over volume therefore prescribes the answer 7/26. What is the prob-
ability of a cube with edge length less then 2cm, then? Is it 1/2 or 7/26?
The principle of indifference seems to say both, and so to contradict
itself.
To catch the indifference principle in this Bertrandian trap, two as-
sumptions were made: that the principle recommends a probability
distribution in any circumstances whatsoever, even given the deepest
ignorance, and that the principle cannot recommend two or more dis-
tinct distributions.
Some supporters of indifference have rejected the first assumption,
holding that the principle can be applied only when a certain unambigu-
ous logical or epistemic structure has been imposed on the problem
(Keynes 1921; Marinoff 1994; Jaynes 2003). Jaynes asks, for example,
with what probability a straw “tossed at random” and landing on a circle

The Logical Way 31
picks out a chord that is longer than the side of an inscribed equilateral
triangle, a question first posed by Bertrand himself. His answer turns on
the problem’s pointed failure to specify the observer’s precise relation to
the circle. From this lack of specification, Jaynes infers that the probabil-
ity distribution over the straw’s position is translationally invariant in
the small, that is, identical for circles that are very close—a conclusion
that he remarks is in any case “suggested by intuition.” This and other
mathematical constraints implied by the problem statement’s meaningful
silences uniquely determine an answer to Bertrand’s question, dissolving
the paradox in one case at least. What is more, the answer corresponds to
the observed frequencies. Contra Ellis, indifference reasoning therefore
allows, Jaynes intimates, prediction of the frequencies“by‘pure thought’”
(Jaynes 2003, 387).2
The second assumption used to generate the Bertrand paradox, that
the indifference principle must supply a unique distribution, can also be
rejected. Suppose that the function of the principle is to provide, not the
truth about a certain objective probability distribution, but a distribution
that is a permissible epistemic starting point given a certain level of un-
certainty. If a range of such starting points are allowed by the canons of
rationality to thinkers in a given state of ignorance—as Bayesians, for
example, typically assume—then a principle specifying what those can-
ons have to say to uninformed reasoners will offer up more than one and
perhaps very many distributions, from among which the reasoner must
freely choose.3
3.3 Splitting Indifference
After Laplace, the notion of probability began to bifurcate. The separa-
tion is now complete. On the one hand, there is physical probability, the
kind of probability that scientific theories ascribe to events or processes
in the world, independently of our epistemic state. Physical probabili-
ties predict and explain frequencies and other statistical patterns. They
are usually understood to include the probabilities attached to gam-
bling setups such as tossed dice and roulette wheels, the probabilities
found in stochastic population genetics—and the probabilities of statis-
tical physics.
On the other hand is epistemic probability. It represents not a state of
the world but something about our attitude to or knowledge of the world,
or alternatively the degree to which one piece of information inductively

1 8 5 9
32
supports another, regardless of whether the information is accurate. Epis-
temic probability is most familiar today in the form of the subjective
probabilities found in Bayesianism and other probabilistic epistemologies;
as such, it represents an attitude toward, or the relative amount of evidence
you have for, a state of affairs’ obtaining.The probabilities that epistemolo-
gists assign to scientific hypotheses constitute one example; another is
constituted by the probabilities you assign to the three doors in section
3.1—they represent something like your current knowledge about, not
the observer-independent facts about, what the doors conceal.
If you want to represent the dynamics of jury deliberations, then you
will use epistemic probability. If you want to calculate advantageous odds
for a gambling game, you will use physical probability. The classical no-
tion was used for both, but the resulting conceptual strain was ruinous.
Physical probability and epistemic probability are now widely agreed to
be two quite different sorts of thing (Cournot 1843; Carnap 1950; Lewis
1980).4
The principle of indifference, which came into the world along with
classical probability, should bifurcate too—so I will argue (following
Strevens 1998 and North 2010). There are two “principles of indiffer-
ence,” then. Each fits Laplace’s schema, in the sense that it takes you
from symmetries and other structural features to probability distribu-
tions. The principles differ in just about every other way: in the kinds of
structure that they take into account, in the kind of probability distribu-
tion they deliver on the basis of the structure, and in the sense of “right-
ness” that they attribute to the distribution in light of the structure.
The first principle of indifference is a physical principle: it leverages
your knowledge of the physical world to provide more knowledge of the
physical world. The probability distributions it delivers are physical prob-
ability distributions; they represent probabilistic facts that are “out there”
and that predict and explain statistical patterns regardless of your grasp
of what is going on. The physical principle derives its distributions from
the physical structure of the systems to whose products the probabilities
are ascribed; use of the principle therefore requires previous knowledge,
or at least surmise, as to the physical workings of the process in ques-
tion. Further, the physical principle aims to deliver the true physical prob-
ability distribution; its application is based on the presumption that the
physical properties that serve as inputs are sufficient, or at least are suffi-
cient often enough in the usual circumstances (whatever they may be), to
determine the actual physical probabilities of the corresponding events.

The Logical Way 33
The paradigmatic application of the physical principle is to a gambling
device such as the tossed dodecahedral die. The physical symmetries of
the die are enough, given certain rather general facts about the workings
of the world, to fix the probability distribution over the faces of the die
(as I will later show); the validity of the physical principle turns on this
connection.
The second principle of indifference is an epistemic principle, not just in
the sense that it tells you how to reason (both principles do that), but in
the sense that its subject matter is right reasoning. It takes as its premise
your epistemic state, with special attention to symmetries in your knowl-
edge, or rather in your ignorance, about a certain set of events. It then de-
livers an epistemic probability distribution over those events that reflects
what you do and do not know about them. The physical indifference
principle is all about the world; the epistemic principle is all about you.
Insofar as the epistemic probabilities it endorses are “correct,” that cor-
rectness does not consist in some correspondence to the world, but rather
in the aptness of the probabilities given your level of ignorance. Either
the probabilities directly represent that state of ignorance, by distributing
themselves so as to make no presuppositions about the world where you
have no grounds to make presuppositions, or they are epistemic probabil-
ities that you can reasonably adopt, given your level of ignorance. (Per-
haps these are two facets of the same notion of what is epistemically ap-
propriate or fitting.)
The paradigmatic application of the epistemic principle is to the case
of the three doors. You have no reason to think that the tiger is behind
one door rather than another. An even allocation of probability among
the doors reflects this state of epistemic balance.
Why think that the case of the doors is governed by a different prin-
ciple than the case of the die? The probability distribution based on the
physical symmetries of the die is a powerful predictive tool, and you
know it: merely by applying the physical principle, you can be confident
that you are in a position to predict patterns of outcomes in die rolls with
great success (if your assumption that the die is well balanced is correct).
The probability distribution based on the epistemic symmetries in the
game of doors does not have this property. Though you know you have
applied the principle correctly, you also know that you should not ex-
pect actual frequencies to match the probabilities so obtained (assuming
for the sake of the argument that you are highly resistant to tiger bites,
so able to play the game many times over).

1 8 5 9
34
Once indifference is split down the middle, and two distinct principles
with separate domains of application are recognized—what I have been
calling the physical principle and the epistemic principle—the objections
surveyed above to reasoning from symmetries and other structures to
probabilities evaporate.
Taking these difficulties in reverse order: the fourth, Bertrand’s para-
dox, should be handled differently by the two principles. The physical
principle will not recommend a probability distribution in cases such as
the cube factory because there is not enough information about the phys-
ics of cube production to go on; for example, there are no known physical
symmetries in the cube-makers.
The epistemic principle may or may not make a recommendation. But
if it does, then I suggest that it should not try to emulate its physical
counterpart by endorsing a unique probability distribution over edge
length, let alone trying to give a distribution that will accurately predict
frequencies (as would-be solvers of Bertrand’s paradox sometimes seem
to be doing). Rather—so my Bayesian instincts tell me—it should sug-
gest a range of possible distributions, proposing that any of them would
be a reasonable probabilistic starting point in the circumstances. Among
the reasonable candidates will be uniform distributions over both edge
length and cube volume. Not included, I imagine, will be a distribution
that puts all its probability on an edge length of 2cm, because such a
distribution is unreasonable given our ignorance of the cube factory’s
workings—or, if you like, it fails to represent our lack of information
about such workings.
Defenders of the principle in its epistemic guise have seldom, I should
acknowledge, taken this line. Pillars of objective Bayesianism such as
Jeffreys (1939) and Jaynes (2003) insist that the principle recommends
a single correct prior probability distribution, or at least a single correct
prior ordering of hypotheses, rather than permitting a range of reason-
able choices. The most recent literature seems also, overwhelmingly, to
suppose that the epistemic principle purports to recommend uniquely
rational distributions (Norton 2008; Novack 2010; White 2010).5
Per-
haps this is, in part, due to a failure to distinguish the two principles, so
demanding from one the uniqueness that is essential only to the mission
of the other.
The third objection was that of Ellis and other empiricists: the use of
the indifference principle by Herschel, the classical probabilists, and their
fellow travelers to gain predictive knowledge of frequencies from igno-

The Logical Way 35
rance alone violates some grand epistemic conservation law. The empiri-
cists are right. There is a principle that recommends probability distribu-
tions on the basis of ignorance alone, and there is a principle that
recommends probability distributions with predictive power. They are
not the same principle.
What are the symmetries, and more generally the structural properties,
on which the indifference principle should base its recommendations?
That is the second problem. Again, the solution depends on which of the
two principles you have in mind. Much of what follows in this book is an
attempt to answer the question for the physical principle. You can at least
see, I hope, that principled answers are possible: the numbers on the face
of the dodecahedral die are not symmetry-breakers because they make
no difference to the physical dynamics of the die.
The first objection to Laplace’s characterization of the indifference
principle was that it seemed simultaneously to be about the world and
your mind, or about facts about equal possibility and facts about your
undecidedness. No wonder; it is two principles twisted into one.6
In the last century or so, the philosophical and allied literature on the
principle of indifference has recognized the dual aspect of the principle
in one way and denied it in another. It has recognized the duality by giv-
ing the principle a more purely epistemic cast than it finds in Laplace and
other classical probabilists. Writers such as Jeffreys (1939) and Jaynes
(2003) emphasize that the principle recommends epistemic probability
distributions on the basis of the user’s epistemic state, which distribu-
tions are supposed to be the best representation of the user’s informa-
tion or lack thereof.
Yet some of the same writers have failed to recognize the existence of
an entirely distinct physical principle. As a consequence, cases such as
the dodecahedral die are treated by both friends and enemies of indiffer-
ence as paradigmatic applications of the epistemic principle. Jaynes’ solu-
tion of Bertrand’s chord problem, for example, fails to countenance the
possibility that our intuitive judgments about the probability distribution’s
symmetries draw on the physical structure of the chord-determination pro-
cess, which would make it less mysterious that the solution matches the
observed frequencies. Jaynes also cites Maxwell’s derivation of the velocity
distribution as a classic application of the epistemic principle to obtain
predictive probabilities. The attribution of these empirical successes to
the epistemic principle clothes it in glory, but ultimately to its great det-
riment, I think, because it generates impossible expectations about the

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—Bien sûr. Quand je suis deux jours sans la voir, cette enfant, j'ai
des vapeurs comme les petites dames. Et pourtant Dieu sait si je
devrais seulement lui ouvrir ma porte! Mais d'abord, laissez-moi me
sécher. Je suis en nage, ma bonne mame Adrien! Un fichu coup de
soleil! Ça prend tout d'un coup! On fond en eau. Mâtin! Il ferait bon
être poisson, ma parole, comme les animaux qui me passent sous la
main.
En voilà qui ont de la chance quand les pêcheurs ne les tracassent
pas! hein! Ce qui va aller ces jours-ci, c'est la limonade! Elle va
gagner des sommes! Ce n'est pas comme nous autres. La marée on
n'en vient pas à bout, d'une pareille chaleur. Et des odeurs, mame
Adrien! Il y a de quoi sentir les maquereaux des buttes Chaumont à
Montrouge. De sacrées affaires, ma pauvre dame!
—Oh! ce n'est pas l'argent qui vous taquine, vous, madame Pivent.
Vos vendanges sont faites. Vous en avez amassé de ces rentes! Vous
voilà à l'abri pour le reste de votre existence. Ce n'est pas comme
moi.
—Ne vous plaignez pas. La loge est bonne. Une fière maison et de
bons bénéfices.
—Euh! il n'y a pas de quoi mettre des mille et des cents de côté à la
fin de l'année et quand on a noué les deux bouts!... Pourtant il y en
a de plus malheureuses que moi et si je n'avais peur de l'avenir...
—Bah! Laissez donc! Il ne faut pas penser aux neiges de décembre
quand on cuit au soleil. Et l'enfant, qu'est-ce que vous en faites?
—Je n'en sais rien. On ne la voit pas souvent.
—Ni moi non plus! C'est-à-dire que je me demande où elle peut
passer tout son temps. Encore, ma pauvre mame Adrien, j'aime
autant ne pas creuser ces choses-là.
La concierge leva les yeux aux chapiteaux des colonnes et ne
répondit pas.

—Voyez-vous, mame Adrien, reprit la poissonnière, il y a des
fatalités. C'est plus fort qu'elle. Elle pouvait être heureuse en vivant
honnêtement avec moi ou même avec un ami. Je lui passerais ça,
car il faut de l'indulgence en ce monde. On n'est pas parfait. Mais
c'est plus fort qu'elle. Tout le sang de son gredin de père! Il faut
qu'elle coure! Et pourtant, voyez-vous, il y a quelque chose qui
m'attire, moi! Elle a des moments où elle est bonne comme défunte
ma pauvre sœur, une brebis du bon Dieu! On ne peut pas la haïr,
moi du moins. Je me jetterais au feu pour elle. Cette gamine-là me
remue quelque chose sous mon corset. Où croyez-vous qu'elle soit,
mame Adrien?
—Elle ne le dit pas.
—Et quand elle le dirait, allez, autant de paroles, autant de couleurs!
La bonne dame tira de sa vaste poitrine un énorme soupir.
—Encore une qui a mal tourné, mame Adrien. Mais ce n'est pas trop
leur faute, à ces jeunesses. D'abord, il y a les hommes, les jolis
cœurs qui leur tournent la tête. Et puis les boutiques, les étalages,
les bijoux, les lingeries, les robes, les figures de cire chez les
coiffeurs avec des perruques! Si ça devrait être permis, ces
tentations-là, ma pauvre dame. Comment voulez-vous qu'elles
résistent! Tenez, voulez-vous mon opinion? Si elle ne vous fait pas
de bien, elle ne vous fera pas de mal. Je suis de l'avis de mon cousin
Méraud. Paris, une sale ville pour les filles! Pas moyen d'y rester
tranquille, à moins d'avoir la tête solide comme votre servante et de
tomber sur un mari comme Pivent, un brave homme, mais ce sont
toujours ceux-là qui partent les premiers, tandis qu'un tas de
vauriens, des propres à rien, ma chère dame, que je pourrais mettre
à mon étalage, ont la vie dure comme des crabes. Ainsi elle n'est
pas là, mais elle se porte bien, dites?
—Très bien, madame Pivent.
—Je vais donc m'en retourner tranquille.

Elle aperçut son panier qu'elle avait oublié.
—Suis-je assez sotte, fit-elle. Cette petite me tournera la tête comme
à mon pauvre homme. Je laisse là dedans ce que je vous apportais,
et par ce temps d'orage!
Elle tira de son panier en jonc, très finement travaillé, une petite
langouste cuite à point et de couleur cardinalesque.
—C'est à votre intention, mame Adrien. Vous êtes d'une pauvre
santé, et pour vous éviter de la peine, ma bonne, Brigitte, l'a mise
dans un court-bouillon de première. C'est frais comme une rose.
Elle s'était levée; elle déposa le crustacé sur une assiette, dans le
salon de la concierge, près de la fenêtre.
—Vous m'en direz des nouvelles quand je reviendrai.
Madame Pivent avait cette qualité qui donne de la grâce au plus laid
des visages. Elle aimait fermement ce qu'elle aimait. Elle était bonne
autant que rude.
Elle tira sa montre, une petite machine microscopique, attachée à
une lourde chaîne très luisante, enroulée autour de son cou.
—Comme le temps passe auprès de vous, mame Adrien, dit-elle.
Cinq heures déjà et je vous fais perdre votre après-midi avec mes
bavardages. Je m'en vais. Je retourne à ma rue du Cygne. Ce n'est
pas beau comme ici, dame non! C'est laid, c'est triste, c'est sombre,
mais je m'y plais; l'habitude! Et je suis toute portée le matin pour la
criée!
La concierge écoutait, parlant peu, par phrases courtes, comme si
elle avait eu peur de se fatiguer.
—Pourquoi y allez-vous? dit-elle. Vous êtes riche.
La marchande de poissons fit claquer sa langue:

—Voilà! Qu'est-ce que je deviendrais? Le temps me durerait, toute
seule. Si encore j'avais ma petite à cajoler. Mais non. Elle ne trouve
pas la maison assez soignée pour elle.
Elle avait remis son panier à son bras et rajusté ses jupes en les
faisant bouffer d'un tour de main.
—Bonsoir, mame Adrien, dit-elle. Ne lui contez pas que je suis
venue! Une ingrate! Je cours prendre l'omnibus dans l'avenue. A la
revue.
Elle s'en alla et la concierge resta seule dans sa maison vide.

XXXVIII
Angèle avait annoncé que la séance serait longue à la Chambre, elle
ne s'était pas trompée. C'était à supposer qu'elle avait consulté une
pythonisse lucide.
L'ordre du jour était chargé de quelques menues affaires telles que
votes d'emprunts ou tarifs de douanes, qui furent expédiées avec
une rapidité vertigineuse.
Mais la grande question était la lutte d'un énergumène des extrêmes
partis contre l'Arpin de la place Beauvau. Tout le Parlement était
sens dessus dessous pour une femme de mœurs faciles, arrêtée
dans l'exercice de ses fonctions.
Il s'agissait de savoir lequel des deux forts tomberait l'autre.
Partout ailleurs le succès de Duvernet n'eût pas été douteux, mais
dans un pays où la foule est toujours du parti du voleur contre le
commissaire, c'était différent. Il fallait voir.
Ce fut une belle bataille.
La tribune trembla sous les coups de poing du champion des
hétaïres à dix francs l'heure et les voûtes du palais retentirent de ses
accents d'ophicléide enrhumé.
Mais il développa ses conclusions avec une prolixité qui compromit
sa cause.

Les estomacs des législateurs demandaient grâce, quand, vers
l'heure du dîner, l'orateur descendit de la tribune en laissant le
champ libre à son adversaire.
Chazolles, étranger à ce qui se passait autour de lui, relisait, au banc
des ministres, le rapport de Melchior Pavie, et une colère effrayante
s'amassait en lui.
Le président du conseil fut bref, incisif et cruel pour la cliente de son
adversaire.
Il démontra qu'elle pratiquait, quoique mariée, une industrie pour
laquelle son conjoint lui laissait les plus larges libertés et dont il
encaissait les recettes.
Un monde intéressant!
Puis prenant les choses de plus haut, il s'éleva contre les
manœuvres de certains êtres hargneux, querelleurs et amis du
trouble, qui jetaient incessamment des cailloux sur les rails du train
gouvernemental, au risque d'amener un déraillement et d'effrayer
nos paisibles populations. Il soutint qu'il fallait aborder les grandes
réformes, un mot magique! travailler utilement sans s'attarder à des
questions oiseuses. Il observa qu'on perdait ainsi un temps précieux
et n'oublia pas d'insinuer que c'était manquer de respect et d'égards
envers des collègues que de les astreindre pour des vétilles, et des
querelles méprisables, à prolonger au delà du nécessaire les séances
déjà trop chargées et à ne trouver à leur retour qu'un de ces repas
flétris par l'auteur de la Gastronomie:
Un dîner réchauffé ne valut jamais rien.
Il fut mordant, hautain et autoritaire, et d'acclamation il enleva un
vote favorable, grâce surtout à l'heure avancée et au vers de
Berchoux.
Mais il était huit heures et demie.
Chazolles se fit conduire chez sa maîtresse.

La femme de chambre causait dans la loge avec la concierge.
—Eh bien?
—Madame est revenue. Elle a changé de toilette; elle est repartie.
Une maîtresse Benoiton!
Chazolles frappa le parquet de sa canne.
—Mais madame a laissé une lettre pour monsieur.
—Où donc?
—Sur le bureau du petit salon. Si monsieur veut...
—Non, j'y vais.
Il monta rapidement à l'appartement d'Angèle.
La lettre l'attendait.
Il la parcourut avec avidité et la rejeta en la froissant à terre.
—Elle se moque de moi, pensa-t-il. C'est clair.
Dans le boudoir et la chambre à coucher, on sentait des odeurs de
jolie femme, de poudre de riz, d'essences légères et discrètes.
Au dehors, la nuit tombait, une belle nuit d'été, claire, argentée par
des lueurs d'étoiles scintillantes dans l'azur sombre et profond.
Affaissé sur un fauteuil bas, Chazolles promenait ses regards,
pendant que ses lèvres exprimaient la désillusion et le dégoût, sur
les tentures de satin du lit, doublées de dentelles crémeuses, sur les
murs chatoyants où, dans la soie et le velours, il avait cru enfermer
et retenir un bonheur qui lui échappait, comme l'oiseau qui sort du
nid dès que ses ailes lui sont poussées.
Il entendit un bruit de voiture s'arrêtant dans la rue.
Son cœur battit avec une violence extrême.

Il y porta ses doigts crispés avec un geste furieux:
—Amour ignoble, pensa-t-il, est-ce que je ne pourrai pas t'arracher
de là?
Il laissa retomber son bras, découragé.
Non, il ne pouvait pas.
Il était contraint de courber la tête et de s'avouer vaincu.
Malgré ce qu'il savait, il se sentait assez lâche pour pardonner
encore si Angèle se jetait à ses genoux.
Il se planta devant un portrait, le seul tableau qui est suspendu aux
murailles capitonnées, et à la lueur d'une bougie qu'il promenait
devant lui, il le considéra longtemps.
Cette toile, un chef-d'œuvre de Carolus Duran, rendait
admirablement le blond bizarre des cheveux à reflets fauves, de ces
cheveux magnifiques qui ruisselaient sur les épaules nues, d'une
blancheur de neige, éclatante comme un rayon de lune.
Les bras minces au poignet se rattachaient à l'épaule par une liaison
harmonieuse; les mains délicates étaient faites pour les caresses.
Le sourire de la bouche, petite et mignonne, et des lèvres de
pourpre, sanglantes, appelait les baisers. Les yeux clairs, d'un bleu
glauque, brillaient sous des sourcils plus foncés que les cheveux.
Il y avait dans l'ensemble, je ne sais quel attrait mystérieux, charnel,
qui la rendait désirable, enivrante, un charme passionnant qui
s'emparait de l'homme, une sorte de volupté tyrannique dont elle
était comme imprégnée et qui grisait en s'infiltrant dans le cœur et
les sens, en dépit de toutes les résistances.
En vérité, elle était de cette beauté insolente, idéale et saisissante
qui fascine et fait commettre les crimes.

Ce n'était pas une femme, c'était la femme dans son incarnation la
plus vraie, dans sa toute-puissante et dominatrice faiblesse.
Le ministre resta abîmé longtemps dans une douloureuse
contemplation.
—Que m'a-t-elle donc fait, dit-il en se redressant, que je ne peux pas
m'en défendre et que je deviens une chose à elle, le jouet de ses
caprices, le complice de ses hontes, une manière de valet à ses
ordres! Ah! je suis trop lâche! Il faut en finir.
Et tout d'un coup, il se souvint qu'il n'avait pas dîné, en se rappelant
la péroraison de son ami Duvernet. C'était un moyen de tuer le
temps.
—Elle me donne rendez-vous à minuit, dit-il; soit, j'y serai.
Il traversa les appartements plongés dans l'obscurité et sortit en
fermant violemment la porte.

XXXIX
Les passants qui arpentaient les trottoirs du faubourg Saint-Honoré
en flânant aux boutiques et qui croisaient ce beau garçon brun,
grand et taillé en hercule, ne se doutaient guère qu'ils avaient
devant eux un des personnages en vue dans les hautes régions du
pouvoir.
Chazolles allait machinalement devant lui, au hasard, comme un
corps sans âme, ou un poète qui poursuit la rime capricieuse et
oublie le monde entier, des nuages où il s'est envolé.
Chazolles ne songeait ni aux passants, ni aux jolies femmes qu'il
frôlait, ni aux palais qui se dressaient à sa droite et à sa gauche.
Son esprit était fixé sur un seul point: cette fille qui avait dérangé sa
vie, et s'était emparée de lui au point de le rendre insensible à tout
ce qui n'était pas elle.
Par quel philtre l'avait-elle enivré? De quelle puissance magique était
donc douée sa prunelle vague et troublante? Quel parfum l'attirait
vers cette chair pâle, pétrie pour le vice et l'orgie?
Il aurait voulu être à cent lieues d'elle, s'enfuir, et il était enchaîné à
sa suite par un lien impossible à rompre, retenu par un aimant
irrésistible et magnétique.
Et il ne se dégagerait pas de cette étreinte mortelle, avilissante!

Il en était arrivé à des confidences de domestiques, à des stations
chez les concierges, à des abaissements inconnus!
A cette idée, il était pris de rage.
Tout à coup, il se trouva à l'angle de la rue Royale, en face du café
Durand brillamment éclairé.
C'était là qu'était mort le baron Germain.
La curiosité le poussant, il entra.
Au dehors, les buveurs de bière étaient nombreux. Des couples
élégants, aux tables de la terrasse, jouissaient, en se rafraîchissant,
de la beauté de cette soirée superbe et de la vue des promeneurs
qui se rendaient aux Champs-Élysées.
La plupart des dîneurs étaient déjà sortis du restaurant.
Quelques-uns seulement achevaient leur repas ou fumaient en
causant.
Par un hasard étrange, il s'assit à la table où Melchior Pavie avait
dîné quelques jours auparavant.
Les garçons s'empressèrent.
Chazolles était de haute mine et de ceux pour lesquels on redouble
de politesse.
Il commanda un dîner banal et se plongea dans la lecture des
journaux du soir.
C'est à peine s'il voyait les lettres s'aligner devant lui.
Sa pensée était vagabonde.
Elle cherchait dans Paris, furetant dans tous les coins et se
demandait où se trouvait Angèle.
L'idée qu'elle se donnait à d'autres lui était insupportable.

Un habitué, qui digérait dans une encoignure, en savourant à petits
coups, de temps à autre, une liqueur qui devait être excellente, à en
juger par ses mines de gourmet ravi, appela le maître d'hôtel, en
habit noir, qui errait dans les salles vides.
L'habitué était un monsieur très bien, aux cheveux gris qui
semblaient poudrés, à la figure pleine, la moustache effilée et cirée
aux extrémités en dards de hérisson.
On aurait dit un marquis Louis XVI descendu de son cadre.
—Vous étiez là l'autre jour, dit-il. Vous avez vu l'accident?
—Oui, monsieur le comte.
—Le baron Germain était de mes connaissances. Je l'avais prévenu.
Il passait les nuits au jeu, courtisait les femmes. Il brûlait la bougie
par les deux bouts. Et la petite femme vous l'avez vue?
—Oui, monsieur le comte.
—Vous avez du goût, Joseph! Vous êtes un connaisseur. Donnez-moi
votre avis. Comment était-elle?
—Ah! monsieur le comte, une ravissante personne! Une bague au
doigt d'un millionnaire!
—En vérité?
—Oui, monsieur le comte. Je ne crois pas qu'il y ait dans Paris une
plus mignonne femme! Des yeux, des dents, des lèvres, des cheveux
surtout! Des cheveux comme il n'y en a pas! Et le reste!
Le maître d'hôtel leva le bras droit avec un petit bruit sifflant qui
s'échappa de sa bouche et valait un poème.
—Vous ne m'étonnez pas, Joseph! Le baron Germain était un expert,
un raffiné. Ce qui me surprend, c'est qu'une si belle fille ait pu
s'accommoder d'un débris pareil. Il craquait de toutes parts. Il devait
s'écrouler.

Le maître d'hôtel eut un sourire fin.
—Monsieur le baron était peut-être très généreux?
—Lui! trop égoïste! un pingre!
—Alors, acheva le maître d'hôtel, c'est que monsieur le baron
achevait les éducations et lançait ses élèves. C'est un métier qui
rapporte.
Chazolles étouffait dans sa peau.
Oh! ce Paris! Quel gouffre et tout son bonheur s'y était englouti.
Hélène, sa femme, s'en était éloignée comme d'une ville de
pestiférés, emmenant ses filles pour les soustraire à l'influence
maligne de l'air qu'on y respire.
Lui, il s'y débattait comme un malheureux enlisé dans les tangues
d'une baie perfide, étouffé par l'eau boueuse qui lui envahit la
bouche.
Pour les autres, il était un favori de la fortune! Pour lui, il n'était
qu'un mari justement odieux à sa femme, traître à ses promesses,
renégat de son passé. L'amour d'une coquine roulée dans toutes les
fanges de Paris, le tenait encagé dans cette passion odieuse et
déshonorante comme un criminel attaché au pilori.
Un flot de dégoût lui montait à la gorge. Et cependant il n'avait
encore, en dépit de la dénonciation flagrante qu'il tenait à la main,
malgré les mille preuves qui éclataient autour de lui comme des
bombes de dynamite et réduisaient en pièces ses croyances et ses
illusions imbéciles, qu'une seule volonté: la revoir; qu'un seul désir:
l'entendre confesser, avec des cris d'effarement, les quelques
légèretés que la malignité du monde transformait en trahisons
grossières et sans excuse.
L'habitué avait fini par se lever, prendre son chapeau, endosser son
pardessus gris en homme méthodique et qui redoute les fraîcheurs

des soirs d'été. Il se dirigea vers la porte non sans adresser le salut
de connaissance à la gracieuse patronne qui siégeait à la caisse.
Chazolles, resté seul, imita l'homme aux cheveux poudrés et à la
moustache pointue, prit son chapeau et suivit l'habitué.
Sur le boulevard, après avoir fait quelques pas au hasard, ne
sachant où se diriger ni comment se distraire jusqu'à minuit, il prit
un fiacre et se fit conduire aux Variétés.
C'était une idée.
Peut-être Angèle s'y trouvait-elle. Il la surprendrait ou se rendrait
ailleurs jusqu'à ce qu'il l'ait découverte.
Il ignorait ce qu'on jouait, mais que lui importait le spectacle?
Il voulait chercher partout. Il aurait fouillé les théâtres l'un après
l'autre, en brûlant le pavé avec un cocher de bonne volonté, quitte à
payer la rosse fourbue, si une certaine pudeur ne l'avait retenu.
Il était dix heures et demie.
Le deuxième acte de Niniche touchait à sa fin.
Chazolles, indifférent à ce qu'on jouait et aux acteurs en scène, à
Judic, Baron et Dupuis, malgré leur incontestable attraction, sonda
toutes les loges, toutes les baignoires de la lorgnette qu'il emprunta
à l'ouvreuse. Il ne négligea pas un coin et parcourut des yeux le
balcon et les avant-scènes.
Rien.
A l'entr'acte, il fit le tour du foyer, mais inutilement.
Angèle n'était pas là.
Il sortit rapidement, courut aux Nouveautés et de là au Vaudeville,
où il offrit aux caissiers le spectacle inouï d'un curieux qui prend son
billet au moment précis où le rideau tombe sur des amants dont les

feux ont été traversés par trois actes de contrariétés et qui vont
célébrer leur mariage dans les coulisses, à la satisfaction du public
qui s'écoule.
Là, il recommença son manège de mari jaloux.
Mais ce fut aussi vainement qu'ailleurs.
Pas de robe caroubier, pas de chapeau caroubier, pas de plume
caroubier contournant de splendides cheveux d'or.
C'était désespérant.
Le ministre se rongeait les doigts de colère.
Où était-elle donc? Où?
Ceux qui ont aimé avec passion, avec rage, ne fût-ce qu'un jour,
qu'une heure, peuvent seuls comprendre le point d'exaltation où il
montait par degrés.
C'était jour d'Opéra.
Il lui restait encore un espoir.
Au sortir du Vaudeville, il se trouva sur les degrés du monument de
l'illustre Garnier sans savoir comment il y était venu.
Les premiers groupes commençaient à défiler pour la sortie et à
l'angle gauche de la façade, au coin de la rue Auber, en se tournant,
il aperçut, mais ce fut comme une ombre qui s'efface, une robe d'un
rouge sombre qui s'engouffrait dans un petit coupé.
Il se précipita.
Mais, au même instant le coupé fila vers le boulevard Haussmann;
une main s'abattait sur l'épaule de l'Excellence et une voix se fit
entendre à son oreille.
Cette voix était celle de Duvernet qui disait:

—Enfin! c'est donc toi! Que diable fais-tu là?
Chazolles voulut se dégager en lançant un énergique:
—Laisse-moi donc, imbécile!
Mais l'autre le retint par un pan de sa redingote.
—Imbécile est vif! Où as-tu l'esprit?
Le coupé était loin.
Il fallait prendre son parti.
—La soirée était belle à l'Opéra? dit-il machinalement.
Le président du conseil passa son bras sous celui de son ami.
—Oh! fit-il avec indifférence. Pour le temps! Assez. Du monde. Pas
mal de diplomates! De la finance. Quelques toilettes. Rien
d'extraordinaire. Ah! si! Le petit duc de Charnay, ton ennemi.
Chazolles tressauta.
—Déjà guéri?
—Parfaitement. Tu le regrettes?
—Oui, je voudrais l'avoir laissé sur le carreau.
—Ah çà! mais, cher ami, tu deviens féroce. Je ne te reconnais plus.
—Il était seul? demanda Chazolles.
—Je l'ignore. Il m'a paru dans sa baignoire dérober au public
quelques amours nouvelles, mais pas moyen de pénétrer l'obscurité
de cette caverne.
—C'était lui, pensa l'amant d'Angèle. Elle lui donne sa revanche.
—Tu as lu mon factum? dit Duvernet. Il est instructif! hein?
—En effet.

—Tu ne me remercies pas, ingrat?
—Si.
—Vois-tu, mon pauvre Maurice, plus je vais, plus je vois que ceux-là
seuls sont heureux qui ne s'attachent à aucune femme si ce n'est à
la leur, eût-elle de légers défauts, qui vivent en philosophes,
jouissent de la comédie que le monde leur donne, et qui, après avoir
usé de tout, abusé de tout peut-être—c'est notre cas à nous deux...
maintenant!—se renferment dans la sagesse d'une vie calme, libérés
des grandes passions qui troublent tout, contents des petits
bonheurs du foyer et de la famille, entre une femme indulgente, et
des enfants qui prennent leur place peu à peu et les repoussent
dans les espaces inconnus d'où nous venons et où nous retournons
tous, les uns en omnibus, les autres à pied, quelques rares
privilégiés dans une bonne voiture capitonnée et suspendue. Nous
sommes de ceux-là. Ne nous plaignons pas. Bonne nuit. Je vais
écrire une grande lettre de quatre pages à Denise et lui annoncer
ma visite. Nous irons ensemble.
Sans attendre la réponse, il serra la main de Chazolles et s'éloigna.
Il s'en allait à pied par les boulevards, respirant à pleins poumons, la
tête haute, regardant les étoiles qui scintillaient, blanches et
diamantées, dans la voûte profonde, léger comme un homme arrivé
au comble d'un désir et dont les rêves sont réalisés, en se disant
qu'après avoir gravi le Capitole il le descendrait comme les autres,
mais sans blessure, en se ménageant une chute moelleuse sur un lit
étendu à l'avance.
—Pauvre Maurice! pensait-il. Il a eu sa crise, tardive. Elle n'en est
que plus violente. Espérons qu'elle va finir.
Chazolles, dès que son ami se fut éloigné, retomba dans ses rêveries
sombres.
Décidément, cette fille se jouait de lui avec une rare impudence.

Et quel personnage elle lui préférait, à lui, si généreux, si prévenant
pour elle.
—Le duc de Charnay! Un poseur qui ne fait même pas aux femmes
qui se laissent éblouir par son titre, l'honneur de les traiter en
gentilhomme français! Un monsieur auquel on prêtait tous les vices,
qui avait des manies de cosaque et cravachait ses maîtresses! Du
moins la chronique scandaleuse le racontait. Un drôle infatué de sa
personne qu'il orne comme une courtisane de bijoux et de brillants!
Un bellâtre mièvre et musqué qu'il aurait cassé en deux d'un coup
de poing! Un besogneux avec son blason, incapable d'entretenir une
femme et trop heureux de la prendre des mains d'un autre et de
promener à son bras des robes et des dentelles dont il ne paie pas
les notes!
Et c'était ce crevé, l'inventeur de ce mot idiot, le pschutt, que cette
fille adorablement belle—car on ne pouvait nier sa beauté,—lui
préférait, malgré les soins et les mille preuves d'amour dont il
l'accablait.
Il était arrivé au faubourg Saint-Honoré.
Il se rappela l'adresse du duc de Charnay, rue de Berry, à l'angle de
la rue de Ponthieu.
En effet, il avait là un petit hôtel assez mesquin, à deux étages, et
d'un ridicule style néo-grec.
Cet hôtel date du premier empire.
La grande porte était fermée.
Deux fenêtres, éclairées, laissaient passer une lumière adoucie à
travers les stores de gaze.
Évidemment c'était la chambre du duc.
Il demeure seul dans cet hôtel avec trois ou quatre domestiques.

Dans la cour, on entendait un bruit de voitures roulées sur le pavé et
de portes qui se refermaient.
Le cœur de Chazolles se serra.
Il restait là en vedette sur le trottoir opposé, cloué malgré lui sur
l'asphalte au coin d'une porte comme un malfaiteur, examinant cette
clarté qui ne s'éteignait pas.
Il crut distinguer des ombres qui se dessinaient sur les rideaux, une
silhouette de femme, reconnaissable à ses cheveux enroulés en
nattes épaisses.
Angèle, sans doute!
Une sueur froide lui ruisselait des tempes.
Au bout de quelques instants, il eut honte.
Les agents qui se promenaient deux par deux l'observaient avec
méfiance.
De rares passants s'écartaient, prenant le milieu de la chaussée,
comme s'ils avaient redouté une fâcheuse surprise.
Lui, un ministre! Lui Chazolles, le brillant Chazolles, réduit à ce rôle
de rôdeur et d'espion!
Quelle honte!
Il gagna la rue du Colisée, qui est à deux pas, et sonna.
La porte s'ouvrit aussitôt.
La loge de madame Adrien était plongée dans l'obscurité, mais les
deux grands candélabres de la cour restaient allumés toute la nuit.
Il entr'ouvrit la loge doucement:
—C'est moi, dit-il. Soyez sans inquiétude.
Il ne demanda pas de renseignements et s'engagea dans l'escalier.

L'appartement d'Angèle était vide.
Le gaz brûlait dans l'antichambre.

XL
Chazolles laissa les portes ouvertes pour bien entendre les bruits de
la maison, et, arrivé à la chambre de sa maîtresse, il s'arrêta de
nouveau en face du portrait de la jeune fille qui le fixait, animée et
vivante.
C'était bien elle, avec ses traits de vierge, l'expression pleine de
douceur abandonnée, sa grâce lumineuse, ses yeux tendres à demi
éteints dans un spasme de volupté.
Et surtout avec ce demi-sourire d'enfant heureuse à qui la vie ne
jette que des fleurs.
Il l'avait eue, bien à lui, il le croyait, pendant des mois entiers; elle
lui avait inspiré une de ces passions frénétiques pour lesquelles on
sacrifierait tout, père, mère, enfants et amis, et maintenant elle en
avait assez; elle courait les aventures; en ce moment même, elle
était aux mains d'un rival exécré; elle le payait de sa blessure et
réparait de ses mains douces le mal d'un coup d'épée dont elle avait
été la cause!
Ah! si c'était à recommencer!
Comme il ne l'épargnerait pas!
La pendule sonna une heure et demie.

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