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The Suntory and Toyota International Centres for Economics and Related Disciplines
Jevons, Bentham and De MorganAuthor(s): R. D. Collison BlackReviewed work(s):Source: Economica, New Series, Vol. 39, No. 154 (May, 1972), pp. 119-134Published by: Blackwell Publishingon behalf of The London School of Economics and Political Science
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1972]
Jevons, Bentham and De Morgan'
By
R. D.
COLLISON LACK
I
I
should like to begin by
expressingmy thanks to the
Provost and
ProfessorialBoardof University
College,and particularlyo Professor
MarianBowleyof the Department
f PoliticalEconomy, or
the honour
which
heyhavedone me in inviting
me to address ou today.
To lecture
at UniversityCollege,London, would be an honour and a privilegeat
any time-but
to
be askedto
deliver he JevonsCentenaryLecture s a
privilege f a veryspecialkind.Specialprivileges arrywiththem
special
responsibilities nd I am very
sensibleof the particular esponsibility
whichrestson
me, charged s
I
am with
the
task of fitly
commemorating
the centenaryof Jevons's
greatestachievement n Economics n the
College
n
whichhe was proudto profess hat subject.William
Stanley
Jevonshas long beenrecognizedas one of the most original
hinkers
who have contributed
o
the
development
f economicscience,and the
Theory of Political Economy
whichhe published ust a hundredyears
ago
has
equally
been
recognized
s
a
landmark
n
the development
f
the
subject.
Over
the
last half century
t
has therefore
attracted he atten-
tion
of
some of the
most distinguished
conomists
o
that
it is
now not
easy
to
say somethingwhich
s at once
true
and freshabout
the
book or
its author.Nevertheless shall hope to be able
to convinceyou that
there
s still
something
o be learntabout
t
todayby looking
at it froma
specificallyUniversity
Collegeviewpoint.
I can best do this,I think, by beginningrom anoutlineof the life of
Jevons.
n
doing
so there s a
specialdifficulty
which
s
perhaps
best met
by recognizing
t
frankly.
I
cannot
and do
not
pretend
to
know as
much about
the life
and work
of Jevons as do some
members
of
this
audience;
no
doubt there are
many
others
in
it
to
whom the main
outlines
of Jevons'scareerare
familiar,
but
there
are
probably
others
again
to whom
they
are not.
Since a
knowledge
of the
main
facts
of
Jevons's
career
s essential
o an
understanding
f
my
theme,
I
hope
those
to
whom
they
are
already
amiliarwill
bear
with
me
while
I
sketch
themin for the benefitof thoseto whomtheyarenot.2
The life
of William
Stanley
Jevonscould
be characterized
n a
sentence
as
short,
varied
and full of
achievement.
evons
was born
in
1835,
the
ninth
child of Thomas
Jevons,
a
Liverpool
ron
merchant;
his
mother
was
the
eldest
daughter
of
William
Roscoe,
a
Liverpool
banker
who
established
high reputation
s
historianand
art
collector.Jevonswas
1
The Jevons
Centenary
Lecture delivered
at University College,
London,
November
3,
1971.
2
For fuller
details
see
Rosamond
Konekamp,
"WilliamStanley
Jevons
(1835-
1882),SomeBiographicalNotes", ManchesterSchoolvol. XXX (1962),pp.251-73.
119
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120
ECONOMICA
[MAY
thus
brought
up
in that circle
of well-to-do
non-conformists
who
contributed
as
much
to
the
intellectual
life
as to the economic
develop-
ment of the north of England in the early Victorian period through their
remarkable
combination
of business
and cultural
interests.
The interest
which these people
had
in
education
is well known,
and
in this company
it
is
hardly
necessary
to stress that University College,
London,
was founded largely
in
response
to the demand
of
middle-class
non-conformists
for
the
higher
education
for their children which
they
were prevented
from obtaining
at Oxford
and Cambridge.
Jevons
himself
was
educated
in
Liverpool
until
the
age
of
15, partly
at
the
Mechanics
Institute
School
whose headmaster
was W.
B.
Hodgson,
later to become the holder of the Chair of Political Economy at the
University
of
Edinburgh.
In 1850
Jevons entered
University College
School,
spending
one
year
as
a
pupil
and
a
further
two as
a
student
at
University College.
In 1853he had decided
to leave
College
and
go
into
business
in
Liverpool,
partly
at least
in
consequence
of the
recurrent
financial
difficulties
which
had
plagued
his
family
since his
father's firm
had
become
bankrupt
as
a
result
of the
railway
crisis of 1847. In
June
1853,
largely through
the
good
offices
of Thomas
Graham,
then the
Professor
of
Chemistry
at
University College,
the
seventeen-year-old
Jevons
was
offered
the
post
of
Assayer
to
the
newly
formed
branch
of
the
Royal
Mint
at
Sydney.
He did
not in fact
leave for
Sydney
until
June 1854,
because
of
delays
in
completing
the
new
Mint
Buildings.
In
the interval
he
had trained
with Graham as
an
Assayer
and taken
a
further diploma
course
on the
subject
in
Paris.
Jevons
remained
in
Australia
until
early
in
1859.
During
most of this
period,
apart
from
his
professional
work in
Chemistry
at
the
Mint, his
interest was
mainly
in the field of the natural sciences such
as
Botany,
and especially Meteorology, but he was also a keen observer of the
rapidly changing
social
scene
in the
Australia of
his
time. The
interest in
social
subjects-particularly
the social life of
towns-which he
had
already
shown
during
his
early years
in
London
seems
to
have
grown
strongly
after
he had been
living
in
Sydney
for
a
year
or
two.
In
1856
and
1857
he
was
supplementing
his
reading
of
Mayhew's
Great
World
of
London
with such
standard
economic works as those of
Adam
Smith
and
John
Stuart Mill and
with
lesser-known
works like
Lardner's
Railway Economy. By
the
beginning
of
1858 Jevons
seems
to have
come
firmlyto the decision to devote his career to the study of social science.
He
wrote
to
his
sister
at this time: "There
are
plenty
of
people engaged
with
physical science,
and
practical
science and the arts
may
be left
to
look after
themselves,
but
thoroughly
to
understand the
principles of
society appears
to
me
now
the most
cogent
business."1
I
W. S.
Jevons to Henrietta
Jevons,
Letters
and Journal
of
W.
Stanley
Jevons,
1886, p.
101
(hereafter
LJ). Jevons recordeddetails
of his
reading
in diaries which
he kept
in
Australia,
but unfortunately
he did not continue this
practice
after
his
return to London.
Relevant
extracts
from
these
diaries
are
included in
the
forth-
coming edition of the Papersand Correspondencef W. S. Jevons, sponsored by
the
Royal Economic
Society.
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1972] JEVONS,
BENTHAM
AND DE
MORGAN
121
Soon after,
Jevons
made
the
decision
to
give up his lucrative post
at the
Sydney
Mint and return to
London,
his
first
objective being to equip
himself for his new career by completing his degree at
University
College. Not unnaturally,
the
decision came as a shock to his
relatives,
for
Jevons's
father had died
without restoring the family
fortunes, and
he and his brothers and sisters had to fend for themselves in the
world.
Nevertheless,
Jevons was convinced that
the
decision was for him
a right
and
necessary
one and he did not
lack the courage to go through
with it.
Resuming
his studies at
University College
in the
autumn of 1859 he
remained in London until 1863, gaining his BA degree in October
1860
and
his
MA
(with
Gold
Medal)
in
June 1862.
This was the period when Jevons "struck out the true theory of
economy"
and suffered the "sad
reverse"
of
being placed fourth instead
of
first in
the
College
examinations
in
Political
Economy because of
what
he
described as
a
"difference
of
opinion
.
.. perfectly allowable"
between himself and
the
Professor,
Jacob
Waley.
Much
has
been written of late about
the
"professionalization"
of
the
study
of
economics,
and
certainly
this had not
proceeded
far in
England
in the
early 1860s,
so that
the path
to
advancement
for a
bright
young
man interested
in
social science was not so clearly marked then
as
it
is
now. After toying with the idea of earning his living by journalism or
other
forms of
writing,
Jevons
fell in
with a
suggestion
made
by
his
cousin Harry
Roscoe that he
should
come to the then
recently-formed
Owens
College, Manchester,
as
a tutor-the
only
tutor in the
College
and
one
who was therefore
expected
to
offer tuition
on
any subject
which
students
could take there. Jevons
did all
that could
be
expected
of
a
struggling young academic,
and
more;
the
work
on The
Coal
Question,
which
he
began
in
1864
and
published
in
1865,
earned him
a national
reputation in a remarkably short space of time-which was probably
not
without
effect in
securing
his
appointment
to the
Chair
of
Logic,
Mental
and
Moral
Philosophy
and Political
Economy
at
Owens
College
in
1866.
It was
in
Manchester
that much of
Jevons's best
creative
work was
done
and
in
the
following
ten
years
his
reputation
rose
from
being
a
national
to an
international one. He became
influential
in
the work of
the
London
(now Royal)
Statistical
Society, gave
evidence before
many
select
committees and
royal
commissions
on
monetary
and other
questions, and was consulted by Robert Lowe, the Chancellor of the
Exchequer,
on
matters
of economic
policy.
His
work
on
logic
and his
pioneering development
of a
logical machine,
ancestor
of
the
modern
computer,
led to
his
election to
Fellowship
of
the
Royal Society
in
1872.2
But the strain
of
combining
all
these activities with
the
heavy
1
LJ, p. 154.
2
See
Jevons,
"On the Mechanical Performance
of Logical
Inference",Philo-
sophical
Transactionisf
the
Royal
Society,
vol.
160
(1870),
pp. 497-518. Jevons's
election to Fellowship is recordedin Proceedingsof the Royal Society, vol. XX
(1872),
p. 198.
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122
ECONOMICA
[MIAY
teaching programme
which
included
much
evening
lecturing
told on
Jevons's
health.
This,
combined
with
the desire to have access
to the
London libraries and the
London circles
in which
he was now a res-
pected authority, led Jevons to decide to return to University College,
this time
as Professor of Political
Economy, a post which he held
from
1876 until
1880.
The
duties
of the
professorship
in those
days
were light; only
one class
was involved,
which met once a week. Indeed, the appointment
carried
less
reward
and less
prestige
than others
which Jevons might
have
taken
-at
Edinburgh
for
example.
But he was anxious
to have a
post
which
would enable him
to
devote
most of his
time to his writing and
the
prospect of a return to his old College was also attractive to him. In
1880,
however,
he
regretfully
decided
that his
strength
was
not
equal
to
a combination
of
teaching
and
writing,
and that
the
many
works which
he wished
to
complete
must have
his
whole attention;
he
therefore
resigned
the
Chair.
Unhappily,
little
time
was
left to
him for the
completion
of
the
projects
which
he had
planned,
especially
the
grand
project
of a
comprehensive
treatise
on
the
principles
of economics.
When his
active
career as an
academic
economist
and
logician
had
lasted
no more
than
twenty years,
he lost his
life
in a drowning accident
in August 1882.
This is no
more than
a
bald
sketch
of a
fascinating life;
but
it should
make clear
that
in that life
University
College, London, was
a
recurrent
motif.
It
would
be
interesting
and
worthwhile
to
expand
on that
theme;
but
in
this lecture
I
wish
to
deal only with a special aspect
of
it-to
consider
two major
intellectual
influences
in
the
development
of
that
economic thought
which
is
especially
typified
in
Jevons's
Theory of
Political
Economy
of 1871. These
intellectual
influences are particularly
connected
with
the history
of
University
College, London, and
I
think
it can be argued that
they came to bear
on Jevons through his connec-
tion with
University
College.
It has
often
been
said
that the Theory of
Political Economy
marks
a
watershed
in
the development
of economic
thought
mainly
because
of
two outstanding characteristics
in
it-its
introduction into economics
of
psychological
hedonism
on the one
hand,
and of mathematical
and
quantitative
techniques
on the other.
The first of
these,
it
seems to
me,
can be
directly traced
to
the utilitarian
philosophy
of
Jeremy
Bentham
and the second to
the mathematical
logic of Augustus De Morgan.
II
It
has long
been a
commonplace
of text-books
in the history of econo-
mic
thought to say
that the
"simple
hedonism" of Bentham
was the
basis of
Jevons's
essentially
subjectivetheory of value.
Like all
stereo-
typed statements,
this is
a half-truth
which merely
skates over the
surface
of
the relationship between
the
thought of
Bentham and
that
of Jevons.
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BENTHAMAND
DE
MORGAN
123
The
fact is
that, as
one
recent writer
has
pungently said, "When
the
concept
of
utility entered economic theory
many
men of
substance
in
the
field were
believing
utilitarians.
Long
before the notion came under
final attack from Sir John Hicks and Sir Roy Allen in the thirties, the
intellectual temper of the times had changed so that
you
could
hardly
have
found a
utilitarian under
a
flat stone."'- It
is
not
surprising then,
with
so
many
economists
trying,
as
Dennis
Robertson
put it,
"to
remove the offending odour
of
utility"2 to find that
modern
commentat-
ors
have
been
disposed
to
minimize
the extent of Jevons's Utilitarianism.
Ross
Robertson,
for
example,
has said that
"one has
the
feeling
as he
reads Chapter
II
of the Theory, the chapter
on
pleasures
and
pains, that
Jevons
is
simply going through
the motions of
citing
an
unquestioned
authority
before
proceeding
to an
altogether
different
kind an
analysis
".3
To
my
mind
this illustrates
an
understandable,
but
regrettable,
vice of
the intellectual historian-that
of
trying
to
explain
the
thinking
of his
subject,
not
in
relation
to the
thought
of the
subject's
own
time,
but in
relation to the ideas of the present time.
In
the Theory of Political Economy Jevons does
in
fact
say
"I
have no
hesitation
in
accepting the Utilitarian theory of morals" ;4
and
to gloss
over
this
does,
I
think,
lead to a fundamental
misconception
of the
natureof the work. If, as I shall try to show, Bentham's ideas permeated
Jevons's Theory inescapably,
then it
is worth while
to
ask, first, how
Jevons
came
under their influence.
A considerable
part
of the
process
of
Jevons's intellectual
develop-
ment
can be
reconstructed
from the
letters
and
diaries which he
wrote
during
his
early life.
In his first
two years as
a
student
at
University
College (1851-53) although
he "had
several
rather
learned discussions
with
Harry [Roscoe]
about
moral
philosophy"5
he
does
not
appear
to
have studied the
subject formally
at
all.
During
his
latter years
in
Australia, when he was beginning to read political economy, he does
not seem to have paid much attention to
philosophy,
and
none at
all
to Bentham.
On his return
to London
and
to
University College
in
the
session of
1859-60
Jevons
did, however,
enrol
in
the
course
on
Philosophy
of
Mind
and
Logic given by John Hoppus-which included
a
section on the hist-
ory
of
philosophy, taking
in
Bentham.6 It seems
unlikely
that
Jevons
would
have derived a
whole new
direction in his
thinking
from
Hoppus,
1
V. C. Walsh, Introduction to ContemporaryMicroeconomics,New York,
1970, p. 24. Of recent years,
there has been a revival of interest in utilitarianism
among philosphers;
cf.
J. Narveson, Morality and Utility,Baltimore, 1967.
2
D. H. Robertson,
Lectureson EconomicPrinciples,vol. 1, 1957,
p. 86.
3
Ross M. Robertson,
"Jevons and his Precursors",
Econometrica, vol. 19
(1951), p. 233.
4 Theoryof Political
Economy(hereafterTPE),
4th
ed., p. 23.
5LJ, p.
23.
6
Calendar
of UniversityCollege
London
for
the session
1859-60, pp.
18-19.
Jevons'senrolment
n the
class is
recorded
n
the
Professor's
Fees
Books, 1859-60.
I
am indebted to Mrs. J.
Percival,
Archivist
of University College, London, for
assistance
n
locating
this
information,
and other details
relating
to De
Morgan's
classesandJevons'sattendanceat them.
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124
ECONOMICA [MAY
whose
lectures
were
notoriously
dull
and
ill-attended.' On
the
other
hand,
in
1860
Jevons
did
go
to "Mr. Martineau's
mental
philosophy
class in
Manchester New
College,
which
is close at
hand in
University
Hall
".2
Now
Martineau was a considerable
philosopher
whose
long
and
deep study
of ethics
had
led
him
to
a
sophisticated
understanding
of
Utilitarianism,
and it seems much more
likely
that
his teaching
would have stimulated Jevons
and
increased
his interest
n
Benthamism.3
There
are various indications from
Jevons's
correspondence
at
this
time that
he
felt
a
growing
interest
in
philosophical
questions,
and
moral
philosophy
in
particular.4
In
the
circumstances it would have
been
surprising
f
Jevons
had not
been
influenced towards
Utilitarianism,
for
apart from the generalacceptabilityof the doctrineat the time therewere
special
factors which would
naturally
have drawn him in that
direction.
Apart
from the obvious and
natural
attraction of the
great
name
of
Bentham
for
any University
College
man,
there
was
also
the
fact
that
Jevons, like
James
Martineau,
was
a
Unitarian. There had
long existed
a
connection
between
Unitarianism in
theology
and
Utilitarianism
in
philosophy and, as has been
remarked by Professor
Mineka, "Unitar-
ians and
Utilitarians
came
largely
from
the same levels of
society, the
'1H. Hale Bellot,
University College, London, 1826-1926, p.
111: "Afterhis first
or second lecture he seldom had
a
pupil;
because, burying
his
face
in his
manu-
script,
he
mumbled so
that
only
an acute
ear could catch
much
of
what he
said,
and
those
who
caught
something
called
it
rot."
(Quoted
from a
manuscript
by
J.
B.
Benson.)
I
LJ, p. 155.
3The
implication
of this view is
that Jevons must
initially
have
become
acquain-
ted with
Bentham's ideas
through
Hoppus's
course, or
the
reading
prescribed
for
it, combined
this
with
his
notions
of
a
mathematical
theory
of
economy, and
subsequently
extended
his
knowledge
and
understanding
of
Utilitarianism
hrough
his
contacts
with
Martineau's
hinking.
For,
according
o
Jevons's
diary
rom
3rd
to
5th February,
1860, he "was
almost entirely
engaged
in
commencing
a
work on
Pol. Econ.... Value to be establishedon the basis of labour and the problems
of
rent,
wages,
interest,
etc. to
be
solved
as mathematical
functions
. .
.". On
February19,
1860 there
is
the
entry:
"At
home
all
day
and
working
chiefly at
Economy,
arriving
as I
suppose
at
a
true
comprehension
of
Value,
regarding
which I
have
lately
very
much blundered."
This,
as Professor
La
Nauze
has
suggested, pinpoints
the
discovery
of
the
utility approach
by
Jevons.
(La
Nauze,
"The
Conception
of Jevons's
Utility
Theory", Economica, vol. XX
(1953), pp.
356-8.)
It was
not
until
October or November
1860 that Jevons
began
to
attend
Martin-
eau's mental
philosophy
lectures,
but
this
does
not
prove
that
Martineau's
eaching
played
no part
in the working
out of Jevons's
ideas, for the
original
manuscript
of the
Brief
Accolnt
of
a
General
Mathematical
Theory of
Political
Economy is
dated "London September27, 1862". Martineau would
certainly have
appealed
to Jevons as one
who,
in
his
own
words,
"carried
nto [moral
and
metaphysical
speculations]
a store of
exclusively
scientific
conceptions,
rendered
amiliar
in
the
elementarystudy
of
mathematics,
mechanics and
chemistry"-Types of
Ethical
Theory,
1885,
Preface,
p.
viii.
Apart
from
the
more
generalpoints
made in
the
text
above,
it
is
interesting
to
note
that the
Brief
Account
contains
one
passage
which
seems
to echo
Martineau as
well
as
Bentham-paragraph
2,
which
begins
with
a
reference to "the
great
springs
of
human
action".
In
1847
Richard
Holt
Hutton,
another
student
of
Martineau
and
later
a
relative of Jevons
by
marriage,
had urged Martineau o
draw
up
"a
graduated
able
of the
springs of
action", and
this
subsequently
became
an
important
part
of
his
lecture
course. Cf. J.
Estlin
Carpenter,
James Martineau,
1906, p. 298.
4
Cf.
LJ, pp.
149,
155-60.
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JEVONS,
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AND DE
MORGAN 125
enterprising,
successful
middle class",
and
shared
inclinations towards
humanitarianism
and
a
faith
in
science.1
This,
as
we
have
seen,
was the
social background
in which
Jevons
had
been reared and
his return to
University College
had
brought
him back
into
it.
Certainly,
however utilitarian philosophy
may
have entered into
Jevons's
thinking,
it
remained
a
part
of
it for the rest of
his
life
and was
much
more
than
just
a
peg
on which
to
hang
the
particular
type of
subjective
analysis
of demand
with
which his Theory
of
Political
Economy
has
come to be
specially
associated.
That his thinking moved
towards
what
Martineau called
"Hedonist
Evolution"
is clear
from the
article
on Mill's Utilitarianism
which
Jevonswrote
for
the Contemporary
Reviewin 1879.2That it remainedfirmlyin the tradition of Bentham is
clear
from
the last book
he
completed
on an economic
subject,
The
State
in Relation
to
Labour,
when
we find
the
statement
that "it
may
be
fearlessly
said
that
no social
transformation
would
be
too
great
to
be
commended
and
attempted
if
only
it could
be
clearly
shown
to
lead
to
the greaterhappiness
of the
community".3
Now,
as
this
quotation
brings
out,
Benthamism was very
much a
social
philosophy;
yet
the
Theory
of
Political
Economy,
with which
we
are
here
primarily
concerned,
deals
very
little
with
social
questions.
How then does Bentham come into it? Clearly it is not the "greatest
happiness
principle",
but the
"principle
of
utility"
which matters
in
the
Theory.
Indeed
that
work
not
only
starts
out
from,
but
revolves
around,
the
theory
of
pleasure
and
pain.4
Lookiing
back
upon this,
later economists
have been
able to
point
out
the
drawbacks
from
which
it suffers
very clearly,
and indeed
they
are
undeniable-it
is
difficult
to evade Marshall's
indictment
that
"Jevons'
great
error was
that of
applying
to
utility
propositions
that
are
only
true
of price".5I shall returnto this point in a moment, but let me suggest
that
if instead
of
looking
back
with the benefit
of
hindsight
we
try
to
put
ourselves
in Jevons's
place
and look
forward,
as
he must
have done
when
first
developing
his
theory (in
1860-62),
the matter
appears
in
a
different
ight.
Already
in 1858 Jevons
had seized
hold of the idea that
"economy,
scientifically
speaking,
is
in fact a sort of
vague
mathematics
which
calculates
the causes
and effects
of man's
industry,
and shows
how
it
may
best
be
applied".6
By
June
1860
he had
"struck
out
... the
true
Theory
of
Economy
...
entirely
mathematical
in
principle
.
.
.".7
His approach,then, was that of a pure scientist; but as a distinguished
I
F. E.
Mineka,
The Dissidence
of Dissent,
Chapel
Hill,
N.C., 1944, p.
146.
2
W.
S.
Jevons,
"John
Stuart
Mill's
Philosophy
Tested;
IV-Utilitarianism",
Contemporary
Review,
vol. XXXVI,
(1879),
pp.
521-38.
3
W.
S.
Jevons,
The State in Relation
to
Labour,
1882,3rd
ed.,
p.
12.
4
I
have
argued
this
point
more
fully
in
my
Introduction
to the Pelican
Classics
edition
of
the
Theoryof
Political
Economy,1970, pp.
15-20.
5
Marshall
to
J.
N. Keynes,
November
26,
1889;
Principles of
Economnics,
th
ed.,
vol. II, p.
260.
6
LJ, p. 101.
7
LJ,
p. 151.
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ECONOMICA
[MAY
philosopher of science, F. S. C.
Northrop, has pointed
out, the subject
matter of
economics, as Jevons and his successors came
to conceive it,
is not objective n the senseeither of the immediately nspected sense
data or the verified, nferred
common-sense objects
.. The question
then
ariseshow
a
deductive
heoryreferring
o such a
subject
mattercan
be empirically erified.... It
might be supposedthat
the verification s
like
that
of
naturalscience,except for the minor difference hat natural
science
appeals to empirical
data given through
the
senses,
whereas
economic
science uses empirical data
given
introspectively....
But
upon
this basis one would
expect
a
different cience
of
economics
for
each
individual.It is clear that
economic
science does not conceive
of
itself
in
this way. It claims
for its theory the same
validity
for
every-
body thatnaturalscienceclaimsfor its laws and verifiedpropositions.
But
how, by appeal to
introspectionrather
than to the
senses,
can one
get
any
criterion
for public validity?1
Now it seems to me
that this
is just the problem which
Jevons,
as
a
young scientist
trying to reason
out "the true Theory of Economy",
must have
encountered-and must have
realized that Bentham
could
provide
an
answer of
satisfying symmetry through the
"principle
of
utility".
Here we must come back to the criticism of Marshall and other later
writers.
Jevons,
in
the
opening pages
of
the
Theory,
says
with
all
the
emphasis
italics
can
give, "our
science
must be
mathematical, simply
because it
deals with quantities".2
In seeking to ground
that
science on
pleasuresand pains which not
only notoriously defy
quantification but
also
differ in
quality was
not
the
scientist being
unscientific,
the
logician
illogical?
Once
again, instead of
trying
to
answer
on our
terms,
let
us
see what answer
Jevons
himself gave-for
he
did
not
overlook
or
evade
these
questions.
First, on the question of higher and lower pleasures, Jevons argued
that
"my
present purpose
is
accomplished
in
pointing
out
this
hierarchy
of
feeling,
and
assigning
a
proper
place
to the
pleasures
and
pains with
which
the
economist deals. It
is
the
lowest
rank
of
feelings
which
we
here
treat".3
So much
is stated
in the
Theory;
but
in
his
critique
of
J. S. Mill's
philosophy
Jevons went
further and indicated
that
the
question
of different
grades
of
pleasure
could
be
logically handled-by
a
strict
adherence to
Bentham's
ideas:
"Nor is it to be
supposed,
that
Bentham, in making his analysis of the coinditions of pleasure, over-
looked the
difference
of
high
and
low;
he
did
not
overlook
it
at
all-he
analysed
it.
A
pleasure
to be
high
must have
the
marks of
intensity,
length, certainty,
fruitfulness
and
purity,
or of some
of
these
at
least;
and
when we
take
Altruism
into
account,
the
feelings
must
be of
wide
extent-that
is,
fruitful
of
pleasure
and
devoid
of
evil to
great
numbers
'
F. S. C.
Northrop, The
Logic of
the
Sciences
and
the
Humanities,
1959
printing,
pp.
240-41.
2 TPE, 4th ed.,
p.
3.
3
TPE,4th ed., pp. 26-7.
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JEVONS, BENTHAM
AND DE MORGAN
127
of
people"; and so, he
argued,
"Mill proposed to give
'geniality'
to
the
Utilitarian
philosophy by
throwing into
confusion what it was
the
very
merit
of
Bentham to
have distinguished
and
arrangedscientifically.
We must hold to the dry old Jeremy, if we are to have any chance of
progress
in
Ethics".'
If by holding to
the dry old
Jeremy
pleasures and pains can all
be
brought
to
a
common
denominator, there
still
remains
the question
in what
units is
it
to
be expressed? To this,
in the first
edition of the
Theory,
Jevons
frankly answered, "Greatly
though I
admire the clear
and
precise notions
of Bentham, I know not
where his
numerical data
are to
be found".2
His
mode of
escape
from
this difficulty was similar
to Marshall's "measuringrod of money"-"A unit of pleasure or of
pain
is difficult even to
conceive;
but
it
is
the amount of
these
feelings
which is
continually
prompting
us
to
buying
and
selling,
borrowing
and
lending,
labouring and resting,
producing
and consuming; and it
is
from the
quantitative
effects
of the
feelings
that
we must
estimate
their
comparative
amounts"'.
Nevertheless,
in
framing
the
rest
of
his
analysis
in terms of
utility
and disutility he
undoubtedly gave
grounds for
Marshall's
criticism4
and
proofs
of
his
allegiance
to Bentham.
III
If, as
seems
undeniable to me, the
influence
of
Bentham in the
creation of the
Theory of Political
Economy
was
considerable and
pervasive,
the
passages
which
I
have
previously quoted
to the effect
that "our
science
must
be mathematical"
reflect
an
influence on
Jevons
which was
earlier and more
profound-that
of
Augustus
De
Morgan.
De Morgan became the first Professor of Mathematics in University
College,
London,
in
1828,
at
the
early age
of
twenty-two.
He
has
been
described
by
its official
historian as
"the
outstanding figure
in
the
first
quarter-century
of the
life
of
the
college",
and
by
the
great chemist
Sir
Henry
Enfield
Roscoe-the
same
Harry
Roscoe
who was
a
cousin
of
Jevons-as "one of the
profoundest
and
subtlest
thinkers of
the
nineteenth
century".
In the fabric
of De
Morgan's
life the
Unitarian thread
again figures
prominently.
He left
Cambridge because of his
objections
to
the
relig-
ious
tests-just
as had
his father-in-law, William
Frend,
himself
a
Second
Wrangler
who
was
deprived
of
his
tutorship
at
Jesus
College.
Frend
was
an
associate
of
many
of the
leading political
and intellectual
figures
of the
early
nineteenth
century,
wrote for
the
Unitarian
Monthly
Repository,
and
was identified with the
earliest
schemes to
set
up
a
1
Contemporary eviewvol. XXXVI, 1879, p. 533.
2
TPE, 1st ed., p. 12.
This passage
does
not appear
in
later editions.
3
TPE, 4th ed., p. 11.
4
Quotedabove, p. 125, n. 5.
5
Bellot, op. cit., pp.
80-81.
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ECONOMICA
[MAY
University
in London free of
all
religious
tests.1
Forty
years
later
it
was
on this
very
issue that De
Morgan resigned
his Chair at
University
College.
In
refusing
to
appoint
James Martineau to
the
Chair
of
Mental
Philosophy in 1866 because he was a Unitarian minister, the College
had, as De
Morgan
saw
it,
betrayed
the
basic
principle
on
which
it
was
founded,
and
he refused
to
have
any
further
connection with
it.2
During
the
thirty
and
more
years
of
his
connection with
University
College,
De
Morgan
made a
substantial
contribution
to
mathematical
knowledge,
but he was also
particularly
interested
in
the
relations
between mathematics and
logic.
Although
he
was
less
successful
here
than his better remembered
contemporary
George
Boole,
he
did
valu-
able work
in
this field. De
Morgan
also
found time
for
an immense
amount
of
semi-popular
writing
on
scientific
subjects,
as well
as
much
public
work,
in
support
of
decimal
coinage
and
on
insurance
matters.3
Nevertheless
it
was
as
a
teacher
rather than a
writer that De
Morgan
enjoyed
his
greatest
success and
reputation.
His own mind
was so
agile
and
so
profound
that
he set a
high
standard for
his
students;
but it
is
clear
from the
testimony
of
many
of
them that
he
possessed
a
rare
gift-
the
capacity
to
convey
to
those
who are
not
naturally
mathematicians
something
of the
significance
of
the
subject,
its
power
and
elegance.4
There is ample evidence of the influencewhich De Morgan had upon
Jevons-it
amounted
almost
to a
fascination.
At the time
of
his matric-
ulation in
1852, Jevons's
main
interests
were in
chemistry and
botany,
but he attended De
Morgan's
"Higher Junior"
classes
and
"Lower
Senior"
Classes.5
By
his
own
testimony
he
found
this
"interesting
though rather
hard";
he
kept up
diligently with
the
class
work
and
formed
the
intention "to
have
all
De
Morgan's
books".6
It
may
seem
that there
is
nothing
very
remarkable in all
this, but
the
effect which
it
had
on
Jevons
comes out
clearly
in
a
letter he
wrote to
his sisterfrom Australia five years later:
For my own
part
I
have never
had the
courage to
open
the many
mathematical ooks I
brought
with
me; but what
do you
think I
would
I
See
Frida
Knight,
University
Rebel,
the
Life of William
Frend,
1757-1841,
1971;
Mineka, op.
cit.,
pp.
148-50;
S. E. De
Morgan,
Memoir
of AuglustusDe
Morgani,
1882, pp.
19-24.
2
Bellot, op.
cit., pp.
339-43;
De
Morgan,
op.
cit., pp.
336-61.
3
De
Morgan,
op.
cit.;
and see also
the
article
"De
Morgan,
Augustus"
which
Jevons
wrote
for
the
Encyclopaedia
Britannica,
11th
ed.,
vol.
VIII,
pp. 8-10.
4
One
historian
of
mathematics
has described
De
Morgan as
"displaying
unusualgiftsas a teacher and scatteringhis energiesrecklessly.His Trigonometry
and Double
Algebra
1849)
contained
certain eatures
of
quaternions,
but
he
did not
follow this
or
any other
theory
to the
conclusion
that
seemed
within his
reach....
His
contributions
to
the
theory
of
probability
still
rank
as
among
the best
in
English and the same
may
be
said
for
his
contributions to
logic....
Had
he
been
able
to confine himself
to one
line,
he
might
have been a
much
greater
though a
less
interestingman." D.
E.
Smith,
History
of
Mathematics,1951
ed.,
vol.
I, p.
462.
5
Shortly
after
this
lecture
was
delivered the
original
manuscript
of
Jevons's
notes
on these
lectures
was quite
unexpectedly
discovered in
the
Library
of the
University
of
Glasgow.
Hence
it
has
become
possible
to fill
in
more of
the
details
of the
mathematical
teaching
which
Jevons
received
from
De
Morgan. See
Appendix,
below.
6
LJ, pp. 22-3. For the seventeen-year-old evons, the intentionmay have been
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JEVONS,
BENTHAMAND
DE MORGAN
129
do if I
had
opportunityever
again?
Attend
College
and De
Morgan's
mathematical
ectures
The
utility of
mathematics s
one of the
most
incomprehensiblehings
about it;
but though I
was
never
bright or
successful in his class, in spite of workinghard, I feel the greatest
benefitfrom
it.1
That there
was
the
greatest
benefit
to
be derived
from
mathematics
by any scientist
was
emphasized
to
Jevons
by
his
cousin
Harry
(Sir
Henry
Enfield)
Roscoe who,
finding
chemistry
to involve
more
mathe-
matics
as
it
advanced,
wished
that he
had
paid
more attention
as
a
student to De
Morgan's
teaching.2
Now
I have
pointed out that
already
in
1858-59 Jevons
had
begun to
feel for those "generalprinciples or laws" which he felt must underlie
the
workings of
society and
to
see them in
terms
of "a
vague sort
of
mathematics". So,
defending
his
decision
to return to
England,
he
wrote to
Harry Roscoe:
"I
wish
especially
to
become
a
good
mathe-
matician,
without which
nothing,
I
am
convinced,
can
be
thoroughly
done. Most of
my
theories
proceed
upon
a
kind of
mathematical
basis,
but I
exceedingly regret
being
unable to
follow
them out
beyond
general
arguments. 3
So the
wish
of
1857 became the fact
of
1860.
"I find
the
classes
at
college a little dull", Jevons wrote to his brotherHerbert, "the charmis
rubbed off
a few
things; but then one learns more and more
to
adore
De
Morgan
as an
unfathomable
fund
of
mathematics".4What
material
did Jevons
draw from
this
unfathomable fund
to
use in
building
his
Theory of Political
Economy?
Perhaps only
some
things
which
lay
near
enough
to the
surface,
in the
shape
of
the elements of
calculus,
it
may
seem.5
more laudable than practical, for De Morgan's text-books are reputed to have
been
"a
mine
of information for the teacher and entirely hopeless for the pupil".
Smith, op. cit.,
vol.
I, p.
462.
1
W.
S. Jevons to
Henrietta Jevons,
June
17, 1857, LJ, pp. 88-9.
2
H. E. Roscoe to W. S. Jevons, February 21, 1854:
"I
mean to go hard at
mathematicswhen I am in Paris.... Nothing can be done without mathematics&
I
hope
to make
up for not attending to
De
Morgan as
I
ought to have done-fool
that
I
was."
3
W.
S.
Jevons to H. E. Roscoe, January,
1859,
LJ, pp. 118-19.
4
W. S. Sevons to Herbert
Sevons,
January27, 1860,
LJ,
p.
150.
5
Therewere other parts of Jevons's work in which other elements drawn from
De
Morgan's "unfathomable und" were important. Perhaps the most notable is
Jevons'suse of probability heory in his interpretationof scientificmethod. Cf. W.
Mays, "Jevons's Conception of Scientific Method",
The
ManchesterSchool, vol.
XXX (1962), p. 226; E. A. Madden, "W. S. Jevonson Induction and
Probability",
in
Blake, Ducasse and Madden,
Theories
of Scientific Method, 1960, p. 247.
In discussion
after
the
delivery
of
this
lecture,
Sir
John
Hicks
drew
my attention
to the
passage
in
Edgeworth'sMathematicalPsychics, 1881, p. 7,
where
he
argues
that the
"implied equatability
of
time-intensity
units"
of
utility
"resembles the
equation
to each other of
undistinguishable
vents or
cases,
which
constitutes
the
first principle of the mathematicalcalculus of belief".
Sir John
suggested that
De
Morgan's
references o
probability
in his lectures
might similarly
have
prompted
Jevons to
think
of utility as a concept susceptible to mathematical treatment,
if
not to
direct
measurement.
I am much indebtedto SirJohnfor this stimulatingsuggestion,whichcertainly
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It is easy
to be
slighting
about
Jevons's mathematics,
as Marshall
was1;
Jevons
himself had no illusions
about his talents
in that
field:
"I am,
of
course,
better
up
to De
Morgan'sbrain-rackings his session",
he wrote in Michaelmas term 1860, "and shall devote much time to
mathematics, yet,
from
having
no
natural talent for
figures
or
quick
memory,
have
no
hope
of
becoming
a
practical
mathematician.
Besides,
it is somewhat
late
in
the
day
at
twenty-six
to learn mathematics, with
which you
will succeed
from the first or never.
"2
But De Morgan
himself drew
a
distinction between
the
analysis
of
the
necessary
Laws
of
Thought,
and the
analysis
of
the
necessary
Matter of
Thought.3
It
was surely
in
the
field of the
laws
of
thought
that Jevons
learnt most from him.
Building
on
the foundations
laid
by
De
Morgan
and Boole,
Jevons
went
on to build
up
his own
system
of
logic,
and contended that
"it
may
be
inferred,
not that
logic
is a
part
of mathematics .
.
.
but
that the
mathematics
are
rather
derivatives of
logic".4
This
statement, which
is
wholly
in the
spirit
of De
Morgan's
work, may serve
to
put the Theory
of
Political
Economy
into its true
perspective
as
a
work
in
mathematical
economics.
The
matter
of
thought
in
it was
not
highly mathematical,
but its whole
structure
was
built up
in
accordance
with a concept
of
scienceas governed by laws of thought whichcould, where quantitiesare
involved,
be
expressed
in
a
mathematical form5;
and
that approach
was
surely
based
on the hard
training
which Jevons
had
received
in
De
Morgan's
classroom.
IV
A commemoration of
a
great work
always runs the risk
of being
either
adulatory
or
patronizing.
On the one hand there is the danger of
seeming to suggest that its author had correctly anticipated everything
derivessupportfrom a passage
n
chapterI
of TPE (lst ed., pp. 9-1
0;
4th ed., p. 8):
"Previous o the
time
of Pascal, who
would have thought of measuringdoubtand
belief? Who could have conceived that the
investigation
of
petty games of chance
would have
led to the
creation
of
perhaps
he
most sublime
branch
of
mathematical
science-the
theory
of
probabilities
?" With the recent
discovery of Jevons's notes
on De
Morgan's
lectures
(see Appendix)
it is now
possible
to
say definitely that
De
Morgan
had
introducedhim
to
elementaryprobabilityconcepts
in his
"Higher
Junior"
class of
1852-53,
and that
Jevons
was
again working on
the
subject, in
moredetail,
n
1860-61,
in
connection with De
Morgan's ectures.Hence, although
the
original manuscript
of the
Brief
Account
contains no direct referenceto these
concepts, it is certainly possible that they played a part in the conception of
Jevons'sutility theory along with what he himself
describedas the "most luminous
and philosophical
view of
existing and possible
systems of symbolic calculus"
which
he found
in
De
Morgan's teaching
and
writing.
1
"They [Cournot
and von
Thtunen]handled
their
mathematics gracefully:
he
seemed
like
David
in
Saul's armour." A. C.
Pigou (ed.), Memorials of Alfred
Marshall,p.
99.
2
W.
S. Jevons to Herbert Jevons,
November 28, 1860,
LJ,
p. 155.
3
M.
J.
M. Hill,
"Some
account of
the
holders of the Chair of Pure Mathe-
matics . .
.", MSS
in
University CollegeLibrary,quoted in Bellot, op. cit., p. 81.
"
W.
S. Jevons to Thomas
Jevons, August
30, 1863, LJ, p.
191.
5
This emerges very clearly in Jevons's statement at the conclusion of TPE,
4th
ed.,
p. 267:
"The
problem
of economics
may,
as it
seems to
me,
be stated
thus:
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131
which
is known about
the
subject
today,
on the
other
that of
pointing
out that we have learned so much since that it seems
hard
to
conceive
how or why
the pioneer originally made
so
many blunders.
I have tried to avoid both these traps by seeking not to make a fresh
evaluation
of
the
Theory of
Political
Economy,
but rather to think
back
into the
structure which Jevons was trying to
build in it
and to
show
where he obtained the chief materials which he used
in
doing
so.
Those
materialswere
peculiarlyconnected with
University College,
but the
work
which
Jevons
chiselled out
of
them
was
peculiarly
his
own.
Whatever
may
have been its
strengths and weaknesses, we come back
to
the fact
which
has brought
us together here today-the
Theory of Political Economy
was a landmark in the development of modern economics, and Jevons
made
it. He was one of many
worthy
students who read Bentham
and profited
from De Morgan's lectures; but, as Foxwell
reminded
Keynes, "the
only point
about Jevons was that
he
was
a
genius".1
Queen's
University,Belfast.
APPENDIX
Oni evons'smathematical tudiesat UniversityCollege,London.
The comments
which
I
made n this lectureon theinfluenceof De
Morgan
on
Jevonswerebased on evidencedrawn rom
Jevons'spersonal ournal
and
correspondence,
and
the
Calendars
of
University College, London, in
addition to the
other printedsources given
in
the
text
above. By
a
remark-
able coincidence,
a
considerable
quantity
of
new
manuscript
evidence be-
came
available only
a
few weeks
after the lecture
had
been
delivered.
Mr. R. P.
Sturges, who
is
acting
as Research Assistant on
the
project
for
compiling
a Guide
to
ArchiveSources
n the
History
of Economic
Thought,
has beentravellingall over the BritishIsles in the searchfor suchmaterial.
In
the course
of
a
trip to Scotland
in
late November, 1971,
he
discovered
in
the Libraryof the Universityof
Glasgow, the following three
tems:
(1)
Manuscript notebook with
title-page: "Lectures
on
Mathematics
delivered
n
University College,London, by
ProfessorA.
De
Morgan.
Session 1852-3. W. S. Jevons"
(Ms.
Gen.
483).
(2)
Manuscriptnotebook with title on
spine: "MathematicalTracts, De
Morgan"(Ms.
Gen.
485).
(3) Manuscript notebook with
title-page: "NOTES,
and
EXTRACTS,
concerningLECTURES on the PURE MATHEMATICS,delivered n
University
College, London, by
Augustus
De
Morgan,
Professor etc.
Session
1860-61".
On
facing page:
"W. S.
Jevons, University
College."
(Ms. Gen.
484).
Given, a certain population, with various needs
and
powers
of
production, in
possession of certain lands and other sources of material: required, the mode of
employing their labour which will
maximize
the utility
of the
produce".
This clearly foreshadows the modern presentation of the same problem in
terms of "constrained maximization", and brings out sharply the real character
of Jevons's break
with the classical
tradition.
1 J. M. Keynes, Essays in Biography,1951 ed., p. 307.
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I
have since had
an
opportunity
o examine
hese
manuscripts,
nd
have no
doubt that
all
three
are
in
the
handwriting
of
Jevons.
According
to
the
records
of
Glasgow
UniversityLibrary,
he
notebooks were
bequeathed
by
George AlexanderGibson,who was Professorof Mathematics n Glasgow
University rom
1909
to
1927
and
died
in
1930;
there
is
no
evidence as to
how
the notebooks came into his
possession.
With the
aid
of these
notebooks
t is
possible
to
put
some flesh
on
the
bare
bones
of
the outline
of
Jevons's mathematicalstudies
previously known.
The syllabus
of De
Morgan's
Mathematics
ectures altered but
little
over
theperiod
of
Jevons's irst
and secondattendances
at
them,
and
ran
typically
as
follows:
The Lower Division of the Junior Class is intended for those Pupils who
possess
very
little
previous acquirement.
The
Subjects
read
are,
the
First
Four
Books of
Euclid;
Arithmetic,
and
the Arithmetical
Theory
of
Propor-
tion;
the Sixth Book of
Euclid;
Solid
Geometry; Algebra,
arithmetically
considered,
as
far
as
equations
of the
first and second
degrees.
The
Higher
Division of the Junior Class is intended
for
those
whose
previous
reading
will
enable them to
begin
the
Fifth Book of
Euclid.
The
Subjects
read
are,
the
Fifth
and Sixth Books of
Euclid;
Solid
Geometry;
a
Review of the
Principles
and
Operations
of
Arithmetic;
Algebra;
Plane
Trigonometry;and,
if
time
permit,
the
Conic Sections
geometrically.
The Lower Division of the Senior Class will
comprehend
those
who have
(either
in the
College
or
elsewhere) passed through
the
Subjects
of
the
precedingClass. The Subjectshere read are, SphericalTrigonometry; Conic
Sections;
applications
of
Algebra
to
Geometry; higher
parts
of
Algebra;
Differential and
Integral
Calculus.
The
subjects
read in the
Higher
Division
will
consist of
Developments
of
the
Differential and
Integral Calculus,
to
prepare
he Student
for
the
higherapplications
of
Mathematics o
Physics.
(Calendar
of
UniversityCollege,
Lonidon, 853-54, pp.
7-8).
The Professors Fees
Books of the
College
record
that
Jevons
was en-
rolled for
De
Morgan's
classes in 1851-52
and
1852-53,
and
again
in
1859-60 and
1860-61;
but
as one fee
could
cover several
classes
this does
not establish which class or division of a class he attended in each year.
These details,
which could
previously only
be inferred
from
references in
Letters and
Journal,
can
now be confirmed
precisely
from
the
notebooks.
Thus the
first
notebook
(MS.
Gen.
483)
makes clear
that
Jevons had
attended
only
De
Morgan's
"Lower Junior" class in
1851-52,
before his
matriculation
into
University
College.
The
first
seventeen
pages of the
notebook
contain
undated
notes on the
subjects
of this
class,
but
the
note-
book
is,
in accordance with
its
title-page, mainly
devoted
to the
lectures of
1852-53.
It
makes clear
that
Jevons attended both
the
"Higher
Junior"
and
"Lower
Senior" classes
during
that
academic
year. Apparently he had al-
ready developed
his
lifelong passion
for
economizing
paper-for
he
made
one
notebook suffice for both
classes,
first
filling
the
right
hand
leaves
from
front to
back,
then
turning
the book
upside
down to fill
the
facing leaves in
the
opposite
direction-a
system
which
does
not
make
the
notes
easy to
follow.
The
second notebook
is
clearly
the one which
Jevons used for
what
he
described as
"the
long
job
of
copying
out De
Morgan's
tracts".'
These
were
tracts which
De
Morgan prepared
on
various
aspects
of
mathematics
1
LJ,
p. 23.
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The same interest
n the logic and philosophy
of mathematicsmanifests
itself in the "Higher Junior"
and
"Lower
Senior" courses,
but Jevons's
comment
of October
31, 1852,
seems
hardly surprising:
"We have
just
finishedwhatwe areto do at presentof doublealgebraand series,whichI
think rather
interesting,
hough
hard".' It
must
have
been a
considerable
effort
to
keep up
with the work of
the two classes at the same
time,
but
Jevons appears
o have done
it
conscientiously,
n
spite
of the fact that his
main interestat this time
was in chemistry,and by the
end of the
academic
year
he
had
got
a
good
grounding
in
the
theory
of
equations
and the
differential alculus.Certainly he
result
of
his
work
was
not
discreditable,
although
Jevons's own account of
it was
modest enough:
"Mathematics
was a
much harder
affair,
of
course.
Some time
before
the examination
I
formed some desperateresolutionsas to the place I would get, and I did
work
up
a
little.
I
tried very
hard
in
the
examination,
but spent too much
time on the hard
ones,
and came out fourth".2
When Jevonsreturned
o
UniversityCollege
in
1859,
he must have found
this earlierexperienceof
"Lower Senior" mathematicsvaluable. Neverthe-
less
it must
again have been
a
considerableeffort for
him to deal with the
two Senior Courses simultaneously,
especially
in
view
of the fact that he
was also
taking
courses in Greek, Latin and German,after a six-yearbreak
from
academicwork.3 So naturallyhe would have been
"better up to De
Morgan'sbrainrackings"
when, after taking his BA degree,he re-attended
the "Higher Senior" class in the following academic year
4
and this
probablyaccountsfor the
more eclectic approachof the
1860-61 notebook.
Integration
Rule of
Derivation by Arbogast's
Bernouilli'stheorem
method
General
formula
of
Reduction Curves of Pursuit
Integration of
rational and integral
Cauchy'sTheorem
functions
Method
of
Parameters
Differential
equations
Systems
of
Equations
Homogeneous
functions
Partial
Differential Equations
Convertible
Operations
Curves, Lines, AnglesSingular Points TrigonometricalFunctions
Infinities
Asymptotes-
Infinity
of
Functions
Problems
on
Lines
and
Planes
Discontinuity
Paraboloid
Singular
Values
Factorial Integrals
Method of
Quadratures
1
LJ, p.
23.
2LJ, p.
36.
3Ibid., p.
148.
4
Ibid., p. 155.