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8/10/2019 James K. Bidwell y Bernard K. Lange - Girolamo Cardano. A defense of his character
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GIROLAMO CARDANO: A Defense of His CharacterAuthor(s): JAMES K. BIDWELL and BERNARD K. LANGESource: The Mathematics Teacher, Vol. 64, No. 1 (JANUARY 1971), pp. 25-31Published by: National Council of Teachers of Mathematics
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8/10/2019 James K. Bidwell y Bernard K. Lange - Girolamo Cardano. A defense of his character
http://slidepdf.com/reader/full/james-k-bidwell-y-bernard-k-lange-girolamo-cardano-a-defense-of-his-character 2/8
GIROLAMO
ARDANO:
A
Defense
of His
Character
By
JAMES
.
BIDWELL
and
BERNARD
K.
LANGE
Central
Michigan
University
Mount
Pleasant,
Michigan
IF WHAT he heard is
correct,
Girolamo
Cardano
was
born
much
to
the
dismay
of
his
parents,
who
were
not
married.
They
attempted
several
unsuccessful
abortions
before
his
birth
on
24
September
1501.
Girolamo's
father,
Fazio,
although
he
practiced
law in
Milan,
found
more
time
for
other
interests,
such
as
medicine,
mathematics,
and
occult
lore.
Fazio
was
highly
esteemed
as
a
scholar
by
his
con
temporaries inMilan. He apparently, for
example,
consulted
with Leonardo
da
Vinci
about
flying
machines.
Perhaps
the
similarities
evidenced
be
tween
the
father and
the
son
may
be
at
tributed
in
part
to
the
fact that
Cardano
was
educated
by
his
father
until
the
age
of
nineteen.
At this
time,
with
the
moral
support
of
his
mother,
he
was
finally
per
mitted
to
attend
his father's
alma
mater
in
Pavia
to
study medicine. The school
closed
a
year
later
because
of
war,
and
he
transferred to
the
University
of
Padua.
There
began
the
stormy
life
of
Cardano
that
ended in
1576.
Much
of
this
life
un
folds
as
we
examine
the
character
of
this
remarkable
man.
It
seems
that
nowhere
else
does
one
find
a
man's
character
so
thoroughly
crucified
as
in
the
writings
about
Cardano
in
most
books
on
the
history
of
mathematics
or
science.
In
general,
students
of
the
history
of
mathematics
consider
him
to
have
been
close to
insanity,
extremely
changeable
in
his
interests,
and
a
man
of
questionable
morals,
to
say
the
least.
Barnes
is
quite
succinct:
Girolamo
Cardan
(1501-1576),
genius
and
charlatan
combined,
pirated
from
Tartaglia
the
solution
of
the cubic
equation
.
. .
, 1
H. W. Turnbull shows much more bias
when
he
states:
Girolamo
Cardan
(1501-1576)
was a
turbu
lent
man
of
genius,
very
unscrupulous,
very
indiscreet,
but
of
commanding
mathematical
ability.
...
He
was
interested
one
day
to
find
that
Tartaglia
held
a
solution of
the
cubic
equation.
Cardan
begged
to
be
told
the
details,
and
eventually
under
a
pledge
of
secrecy
ob
tained
what
he
wanted.
Then he
calmly
pro
ceeded
to
publish
it as
his
own
unaided
work.
.
.
.
He
seems
to
have been
equally
un
generous
with
the
treatment
of
his
pupil
Ferrari.
.
.
,2
As Ball
puts
it:
A
gambler,
if
not
a
murderer
...
at
one
time
in
his
life
he
was
devoted
to
intrigues
which
were
a
scandal
even
in
the
sixteenth
century,
at
another
he
did
nothing
but
rave
on
astrology,
and
yet
at
another
he
declared
that
philosophy
was
the
only
subject
worthy
of
man's
attention.
He
was
the
genius
that
was
closely
allied
to
madness.3
A
more
conservative
example
of
the
sensationalism
directed
against
Cardano
is
Smith's
description
(we
have
added
the
numbering)
:
Girolamo
was a man
of
remarkable
contrasts.
He
was
[l
]
an
astrologer
and
yet
a
serious
stu
1.
H.
E.
Barnes,
An
Intellectuel
and
Cultural His
tory
of
the
Western
World
(New
York:
Dover
Publica
tions,
1965),
p.
569.
2.
H.
W.
Turnbull,
The
Great
Mathematicians,
in
The
World
ofMathematics,
4
vols., ed. J.R. Newman
(New
York:
Simon &
Schuster,
1956),
1: 119.
3. W.
W.
R.
Ball,
A
Short
Account
of
the
History
of
Mathematics,
4th
ed.
(New
York:
Dover
Publica
tions,
1960),
p.
224.
January
1971
25
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8/10/2019 James K. Bidwell y Bernard K. Lange - Girolamo Cardano. A defense of his character
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dent
of
philosophy,
[2]
a
gambler
and
yet
a
first
class
algebraist,
[3]
a
physicist
of accurate
habits of observation
and
yet
a
man
whose
statements
were
extremely
unreliable,
[4]
a
physician
and
yet
the
father
and
defender
of
a
murderer,
[5]
at
one
time
a
professor
in
the
University of Bologna and at another time an
inmate
of
an
almshouse,
[6]
a
victim
of
blind
superstition
and
yet
the
rector
of the
College
of
Physicians
at
Milan,
[7
a
heretic
who
ventured
to
publish
the
horoscope
of
Christ
and
yet
the
recipient
of
a
pension
from
the
Pope,
[8]
always
a man
of
extremes,
always
a man
of
genius,
always
a
man
devoid
of
principle.4
One
must
admit
that
the contrasts
Smith
presents
to
his
readers
are
indeed
remarkable,
but
only
in
the
light
of
Smith's twentieth-century values and his
apparent
lack
of
knowledge
of
sixteenth
century
man.
All of
the
commentators
quoted
seem
to
have
little
knowledge
of
the times in
which
Cardano
lived.
Since
Smith is
considered
by
most of
his
con
temporaries
a
first-rate
scholar
in
the
field
of
history
of
mathematics,
it
seems
ap
propriate,
within
the
scope
of
this
article,
to
investigate
the
nature of
Cardano
by
using Smith's description of him as our
guideline.
Accordingly,
let
us
review
the
contrasts
in
the
quotation
above
in
num
erical
order.
1.
That
Cardano
was
an
astrologer
and
yet
a
serious
student
of
philosophy
does
not
seem
to
be such
a
remarkable
contrast.
In
Cardano's time
there
was
no
stigma
attached
to
astrology.
In
fact,
it
was
expected
that
anyone
who
was
learned
in
mathematics
and
astronomy
would also
be
an
astrologer.
According
to
Will
Durant,
most
of
the
governments
had
court
astrologers
and
many
university
professors
issued
predictions
based
on
astrology.5
Since
astrology
and
mathe
matics
were so
interconnected
in
the
work
ings
of
sixteenth-century
society,
Smith
could
just
as
well
have
said,
He
was
a
mathematician and
yet
a
serious
student
of
philosophy.
2.
In
stating
the
second
remarkable
contrast,
Smith
seems
to
contend
that
first-class
mathematicians
by
definition
should
not
be
gamblers.
Indeed,
modern
mathematicians
are
not
usually
inveterate
crapshooters. However, in the sixteenth
century, games,
including chess,
were
played
for
money.
They
were
the domi
nant
form
of
recreation
at
that
time.6 In
the
autobiography
written
at
the
end of
his
life,
Cardano
confesses
to
being
a
gambler
in
his
early
years.
If,
nevertheless,
anyone may
wish
to
rise
in
my
defense,
let him
not
say
that
I
had
any
love
for
gambling,
but
rather that
I
loathed
the
necessities
which
goaded
me
to
gambling?cal
umnies,
injustices,
poverty,
the
contemptuous
behavior of
certain
men,
the
lack of
organiza
tion
in
my
affairs,
the
realization
that
I
was
de
spised
by
many,
my
own
morbid
nature,
and
finally
the
graceless
idleness which
sprang
from
all
these.
It is
a
proof
of
the
foregoing
assertion
that
once
I
was
privileged
to
act
a
respectable
part
in
life,
I
abandoned
those
low
diversions.
Accordingly
it
was
not
a
love of
gambling,
not
a
taste
for riotous
living
which
lured
me,
but
the
odium
of
my
estate
and
a
desire
to
escape,
which
compelled
me.7
Because of
the death
of
his
father,
gam
bling
became
Cardano's
main
source
of in
come
while
a
college
student.
His
work in
algebra
came
in
his
middle
years.
Even in
his
early
years
he
could
treat
gambling
objectively
enough
to
write
an
early
draft
of
his famous book
On
Games
of
Chance.
3.
It is
probable
that
Smith is
in
factual
error
when
he
says
that
Cardano
was
a
physicist of accurate habits of observa
tion.
Cardano
lists
forty
books
under
Physics
in
his
list of
published
works.
Twenty-two
of
these
are
entitled
On
Subtlety.
His
writings
merely
illustrate
the
state
of
natural
science in
his
time,
and
much
of
his
work
is
based
on
open
specula
tion
or
secondhand
information
rather
than
on
his
own
experiences.
We
believe
that
Cardano
could be
described
accu
4.
D.
E.
Smith,
History
of
Mathematics,
2
vols.
(New
York:
Dover
Publications,
1958),
1:
296.
5.
Will
Durant,
The
Renaissance
(New
York:
Simon
&
Schuster, 1953),
p.
528.
6. Oystein Ore, Cardano, the Gambling Scholar
(Princeton,
N.J.:
Princeton
University
Press,
1953),
p.
178.
7.
Jerome
Cardan,
The Book
of
My Life,
trans.
Jean
Stoner
(London
:
J.
M.
Dent
&
Sons, 1931),
p.
73.
26
The
Mathematics
Teacher
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8/10/2019 James K. Bidwell y Bernard K. Lange - Girolamo Cardano. A defense of his character
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rately
as a
writer
of
popular
science.
It
must
be
admitted
that
Cardano
did make
unreliable statements
on
many
topics
as
seen
in
the
light
of
modern
judgments.
However,
he
was
one
of
the most
produc
tive writers, in terms of quantity, of the
sixteenth
century.
4.
The
next
contrast
seems
trivial.
Cardano
was,
even
by
twentieth-century
standards,
a
physician.
He
regarded
him
self
primarily
as
one,
and
he
devoted
one
chapter
of
his
Life
to
a
list of
his
successes
as a
physician.
In
fact,
he
was
famous
in
his
own
day
as one
of
the
two
greatest
physicians
in
all
of
Europe.
He
traveled
widely to treat well-known persons. His
fellow
practitioners
looked
at
him
in
one
of two
ways
:
either
he
was an
unorthodox,
meddling
fool,
or
he
was
an
initiator
and
a
leader
of
the
reformation
of
medicine.
In
part
because of
his
election
to
the
rec
torship
of
the
College
of
Physicians
at
Milan,
he became
the
latter.
We
are
surprised
to
see
Smith,
writing
in
the
enlightened
twentieth
century,
condemn another man because his son was
a
murderer
regardless
of
the circumstances
of the
crime.
Looking
at
these circum
stances,
one can
understand
why
Cardano
would defend
Giambatista,
his favorite
son.
Cardano
helped
the
son
to
follow
in
his
footsteps
until he earned his medical
degree
at
Pavia.
Giambatista married
a
girl
whom
Cardano considered
highly
disreputable.
After
a
very
stormy
mar
riage
she
chided
her husband
openly
for
not
being
the father of
any
ofhis children.
In
a
fit
of
rage,
Giambatista
poisoned
his
shamelessly
unfaithful
wife. Giambatista
was
executed
when
he
was
not
quite
twenty-six
years
old. His
father
never
re
covered from the
tragedy.
5. The
contrast between
an
inmate
at
an
almshouse and
a
professor
at
the
Uni
versity
of
Bologna
is
another
contrast
that
is
not
so
remarkable
after
it is viewed
in
light
of the circumstances involved.
First
of
all, having
recently
married
at
the
age
of
thirty-one,
Cardano
became
aware
of
the
fact
that,
as
a
country
doctor
in
the
village
of
Sacco,
he
had
no
hope
of
supporting
a
family.
He
had
applied
to
the
Milan
College
of
Physicians
repeatedly
since
moving
to
Sacco
but
had
always
been
turned down. He thenmoved toGallarate,
a
small
town
just
outside
Milan. Here
he
was
closer
to
his
adversaries
as
well
as
closer to
his
acquaintances,
who
might
be
influential
in
getting
him
licensed.
How
ever,
things
were
much
worse
financially
in
Gallarate;
and
he,
his
wife,
and newborn
Giambatista
eventually
took
shelter
in
an
almshouse
in
Milan
for
a
short
time
until,
through
the
acquaintance
of
nobles
in
terested in scientific questions, he was
appointed
public
lecturer
in the
Piatti
Foundation,
a
position
that
his
father
had
held for
years.
About
four
years
after
being
an
in
mate of
an
almshouse,
Cardano
launched
a
violent offensive
against
the
College
of
Physicians
by
publishing
his first
book,
On
the
Bad Practices
of
Medicine in
Com
mon
Use.
Within
three
years,
through
in
fluential
friends
and
sponsors
he
had
gained
by
his
book,
Cardano became
a
full
fledged
member
of the
guild,
and
within
a
few
years
after
that he
was
rector
of the
College
of
Physicians.
About
twenty
years
later
he
became
a
professor
at
the
Uni
versity
of
Bologna.
6.
That
a man
could
be
a
victim of
blind
superstition
and
yet
be
the
rector
of
the
College
of
Physicians
is
definitely
a re
markable
contrast.
On
the
other
hand,
what Smith contends is blind
superstition
would be
regarded
in
the
sixteenth
cen
tury
as
commonplace
beliefs.
As
H.
E.
Barnes
explains:
[Humanist
Science]
collected almost
every
thing
previously
known
about the
occult,
and
is
therefore
a
disheartening
body
of
material to
the
historian
of natural
science.
Even the math
ematical
foundations
of
Platonism
were
bent
to
the
service
of
the
occult.
The
really
important
science
of
this
era was
a
natural and
unbroken
continuation of the later medieval science . . . .8
8.
Barnes,
An
Intellectual
and
Cultural
History,
p.
568.
January
1971
27
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Within his
own
time Cardano
was
not
thought
to be
a
superstitious
fool.
Before
he
was
fifty
years
old Cardano
was
second
only
to
Vesalius
among
European
physi
cians
and
was
overwhelmed
with
flattering
and
magnificent
offers
for
his
services.
The
pope
and
Europe's
royal
and
imperial
heads with their
princely
families
were
convinced
that
no
physi
cian
could better
safeguard
their
health
than
Cardano.9
7. The
high
esteem
inwhich
he
was
held
was
undoubtedly
instrumental
in
his
being
treated with
leniency
by
the
in
quisitor
general
of
the
Counter-Reforma
tion. That
he
was
a
heretic has
never
been
substantiated. The
worst
thing
he
could
have done to bring suspicion on himself
was
to
cast
the
horoscope
of
Christ.
From
information
available,
one
suspects
that
not
only
was
the
horoscope
never
pub
lished
but,
in
fact,
Cardano
never
ser
iously
considered
publishing
it.
It
is
not
known
why
Cardano
was
tried for
impiety.
The
pope
and
his cardinals
were
especially
interested in
making examples
out
of
the
most
popular
authors
of
the
day.
In
his
works
Cardano
made
enough slips
concern
ing
religion
to
qualify
for
prosecution.
His
dedication
of
Ars
magna
to
a
minor
leader
of
the
Reformation
may
have
been
the
cause
of
his
imprisonment.
8.
We feel
that
the
weakest
part
of
Smith's
description
is
the
claim
that
Cardano
was
always
a
man
devoid
of
principle.
This is
self-evidently
a
blanket
statement.
There
are
a
few
portions
of
Cardano's
life
where
his
ethics
can
be
interrogated
authentically.
The most dis
cussed
incident
is
his
dealings
with
Tar
taglia
over
the
solution
of cubic
equations.
Most
of
the
character
sketches
of
Cardano
are
based
on
his
supposed
inhuman and
unprincipled
treatment of
Tartaglia.
We
shall
see
that,
in
fact,
his treatment
was
neither of
these.
Niccolo
Tartaglia
considered
Cardano
to
be
his
only
rival
in
mathematics
worthy
of serious consideration. Cardano is ac
cused
by
modern
writers of
having
ob
tained
from
Tartaglia
a
rule
for
solving
the
cosa
and
cube
equation
after
swearing
an
oath
of
secrecy
and
then
publishing
the
rule
under
his
own
name.10
The
following
brief
account of
the
circumstances
of
the
encounter between these two men and the
actual
discovery
of
the
general
solution
is
based
on
information found in
a
variety
of
sources.
It is
organized
to
present
the
known
events
in
perspective.
The
article
by
Martin
Noorgaard
was
particularly
useful.11
At
the
time
of
the
dispute,
the
prevalent
attitude
in
the
academic
world
was
that
discoveries
of
methods
were
to
be
kept
secret
and
were
considered
to
be
private
property.
This
was
because
scholarly
recognition
was,
for
the
most
part,
based
on
challenge
disputes
or
public
contests.
Not
only
could
considerable
sums
of
money
be
gained
by
the winners
of
such
a
contest,
but also
the
outcomes
of
such
contests
strongly
influenced the
appoint
ment
decisions
of
the
university
senate,
since
most
university appointments
were
temporary
and
subject
to
renewal based
on demonstrated achievement and not on
longevity.
Printing
became
more
commonplace
later in
the
sixteenth
century.
The
attitude
gradually
changed
to
the modern
view
that the
publication
of
his
secrets
is the
scholar's
way
to
recognition.
Thus
we
see
the
inevitability
of
conflict
between
schol
ars
of
medieval
attitude
and
those
of
the
more
modern
viewpoint.
Scipione del Ferro, a professor ofmath
ematics
at
the
University
of
Bologna,
who
had
the
typical
medieval
attitude,
was
evidently
the
first
inventor
of
the
general
rule
used
to
solve
equations
of the
form
xz
+
px
=
q.
His
discovery
took
place
be
tween
1500
and
1515.
Only
two
persons
had
access
to
his
secret.
One,
Annibale
della
Nave,
was
his
son-in-law and
succ?s
9.
Ore,
Cardano,
p.
13.
10.
Cosa
and cube
equations
were
of
the
form
xz
+
px
=
q,
xz
+
q
=
px,
or
xz
=
px + q
in
modern
notation.
11.
Martin
Noorgaard,
Sidelights
on
the
Cardan
Tartaglia
Controversy,
National
Mathematics
Mag
azine
12
(April 1938)
:
327-46.
28
he
Mathematics
Teaetter
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sor
to
his
chair
at
the
University
of
Bologna.
The
other,
Antonio
Maria
Fior,
was
a
rather mediocre
pupil
of
del
Ferro's.
Tartaglia
entered
the
story
in
1530
when
Zuanne de
Tonini da Coi
proposed
prob
lems of the cube and censi type.12 Tar
taglia
solved these
problems
and
also
claimed
that
he
had
the
general
solution
for
this
case.
Fior
heard
of
Tartagliai
claim
and
sought
to
gain
a
reputation
by
challenging
him
to
a
contest.
In
1535
Fior sent
Tartaglia
problems
of
the
cube
and
cosa
type
(which
Fior
knew
how
to
solve),
and
Tartaglia
sent
back
problems
of
the cube
and
censi
type.
Fior
could
not
solve
any
of these
problems.
Tartaglia
struggled
over
the
problems
he
had
to
solve
and discovered
the method
of
solu
tion
just
prior
to
the
time
limit for
the
contest.
Thus
Tartaglia
now
claimed
to
be
able
to
solve both
types
of
cubic
equa
tions.
Da
Coi,
a
friend
of
Cardano,
informed
him of
Tartagliai
work.
Cardano
had
ap
parently
believed
that such
problems
were
not
solvable.
At
the
time he
was
writing
his famous mathematical work Ars magna
(The
Great
Art).
In
1539
Cardano,
through
da
Coi,
attempted
to
persuade
Tartaglia
to
give
him
the
general
solution
for
inclu
sion
in
the
book. Cardano
promised
to
give
Tartaglia
full
credit.
Tartaglia
was
interested
but refused to
cooperate,
since
he
wished
to
write
his
own
book
on
the
subject.
Cardano
persuaded
him
to
reveal
the
secret and
(according
to
Tartaglia)
promised never to publish it.
In
1543
Cardano and
his
pupil,
Lodo
vico
Ferrari,
journeyed
to
Bologna
and
examined the
papers
of
del Ferro.
They
verified
that
del
Ferro
had
first
discovered
the
general
method.
This
apparently
changed
Cardano's
feelings
about the oath
of
no
publication
he
(supposedly)
had
given
to
Tartaglia.
As
Ore
states:
To
the
medieval
mind,
as
one
sees
from
so
many
instances,
an
oath
was
only
valid
in
its
most
literal
sense
and
here
was
cir
cumstance
which
formally
invalidated
Cardano's commitment.
13
Thus
when Ars
magna
appeared
in
1545,
the
formula
was
included
in
chapter
11
on
cosa and cube equations. Tartaglia and
del
Ferro
were
given
their
due
credit. In
fact,
in
chapter
1
of
the
book
Cardano
gives
a
history
of
the
whole affair
outlined
above.
In
our
own
days Scipione
del
Ferro
of
Bologna
has
solved the
case
of
the
cube
and
first
power
equal
to
a
constant,
a
very
elegant
and
admir
able
accomplishment.
...
In
emulation
of
him,
my
friend
Niccolo
Tartaglia
of
Brescia,
wanting
not to
be
outdone,
solved the same case
when
he got into
a
contestwith his [Scipione's] pupil,
Antonio
Maria
Fior,
and,
moved
by
my many
entreaties,
gave
it to
me.
For
I
had
been
de
ceived
by
thewords of
Luca
Pacioli,
who
denied
that
any
more
general
rule
could be
discovered
than
his
own.
.
.
.
Then,
however,
having
re
ceived
Tartagliai
solution
and
seeking
for
a
proof
of
it,
I
came
to
understand that
there
were
a
great many
other
things
that could also be
had.
Pursuing
this
thought
and
with
increased
con
fidence,
I
discovered
these
others,
partly
by
myself
and
partly
through
Lodovico
Ferrari,
formerly
my
pupil.14
The
reader
may
have
considered
that
Cardano
also
fabricated the
discovery
of
del
Ferro's
papers
long
after
his death
in
order
to
prove
his
lack
of
commitment
to
Tartaglia. However,
in
1923
Ettore
Bortollotti
discovered
del
Ferro's
original
papers
in
the
library
of
the
University
of
Bologna.15
The recent
translation
by
Richard
Witmer
of
Ars
magna
into
English
makes
this work of
Cardano's
available
to
the
modern reader. An
examination
of this
work has
convinced
the
authors
that
Cardano
was
indeed
a
first-class
mathe
matician
in
his
day.
All
the
variations
of
cubic
equations
(twenty-two
primitive
cases
are
listed)
are
completely analyzed.
12.
Cube
and
cerni
problems
had
equations
of the
form
zz
+
px2
-
q,
xz
+
q
=
pa?,
or
xz
+
ps2
-}-
q
in
modern
notation.
13.
Ore,
Cardano,
p.
84.
14.
Girolamo
Cardano,
The
Great
Art,
trans.
T.
R.
Witmer
(Cambridge,
Mass.:
M.I.T.
Press,
1968),
pp.
8-9.
15.
Ettore
Bortollotti,
Manoscritti
Matematici
Riguardanti
la Storia
dell'Algebra,
Esistenti
nelle
Biblioteche
di
Bologna,
Esercitazioni
Matematisch?
di
Catania
3
(1923):
81.
January
1971 29
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Geometrie
algebra proofs
are
given
for
each
case,
and rules
for
computation
are
included.
Examples
of
equations
and
prob
lems
yielding
equations
are
given.
The
book
also contains
much
work
on
quartic
equations.
Work with
rules
for
negative
quantities
are
offered,
and
square
roots
of
negatives
are
considered. The
reader
of
The
Great
Art
has
to
be
impressed
with
the
care
and
completeness
Cardano
gives
to
this
book.
As
we
have
seen,
Cardano
very
carefully gives
credit
to those
who
had
solved
cubics and
to
those,
such
as
da
Coi,
who
had
posed
problems.
We
invite the
reader
to
examine this
book himself.
It thus
appears
that
Cardano's
ability
as a mathematician and his honesty as a
scholar
are
certainly
acceptable
to
us
to
day.
Tartaglia
is
usually
viewed
as
the
man
betrayed,
yet
Tartaglia
himself
was
not
a
purist
as a
scholar.
Boyer
notes:
Lest
one
feel
undue
sympathy
for Tar
taglia,
it
may
be
noted that
he
had
pub
lished
an
Archimedean
translation
(1543),
derived
from
Moerbeke,
leaving
the
im
pression
that
it
was
his own. 16
It
may
be that Tartaglia was more the villain.
He
may
have
hoped
to
increase his
reputa
tion
by
attacking
Cardano.
It
is also
in
teresting
to
note
that
Tartaglia
never
wrote
about
his work
in
solving
cubics,
although
he
talked
of
doing
so
for
many
years.
A
second
work
by
Cardano,
which has
been
as
much derated
by
historians
as
Ars
magna
has been
raved
over,
is
his
book
on
gambling,
De
ludo
aleae
(On
Games
of
Chance).
Posthumously
published,
this
book
is
an
expos?
and
a
philosophical
analysis
of
gambling
and
gamblers.
It
con
tains
a
description
of
most
of
the
games
of
his
day,
as
well
as
meditative reflections
on
hunches and methods for
predicting prob
ability.
Cardano
finally
leads
up
to
a
few
well-founded
formulas
on
probability.
For
anyone
interested
in
mathematical
prob
ability,
the
use
of
the
book
is
productive
if he is
willing
to
study
it
seriously.
In
his book
on
Cardano
(which
includes
a
translation
of
On
Games
of
Chance)
Ore
voices the
opinion
that the book is under
rated
simply
because
most
writers have
made
no
attempt
to examine it.
Ore's
claim that
Cardano should
be
ranked with
Pascal
and
Ferm?t
as a
discoverer
of the
field of
mathematical
probability
is
con
vincing.
When
one
takes into
account that
Cardano's
book
represents
the initial effort
n
this
field,
it
must
be
admitted
that it
is
remarkably
success
ful. Even
if his
achievements
had
been limited
only
to
the correct
chances
on
dice,
it
could
have
been
regarded
as
a
great
forward
step,
but
Cardano
goes
much
further.
He
succeeds
in
formulating
certain
fundamental
principles;
he
understands to some extent the law of
large
numbers, and,
after
some
false
starts,
he
is able
to
derive
quite
generally
the
so-called
power
law
for
the
repetition
of events.17
Even
writers
who
have
studied
De
ludo
aleae
more
seriously
have
simply
not
given
Cardano
a
chance.
One
reason
for
this
is that the
book is
inadequately
written
in
four
ways:
1.
The book does
not
offer
a
sufficient
amount
of background information on the games con
tained
in it.
As
a
result it
can
be understood
adequately only
by
reliance
on
related
materials
that
explain
the
games
of the
sixteenth
century.
2.
Cardano's
mathematical
analyses
are
for
the
most
part
obscure
to
modern
man
owing
to
the lack
of
mathematical
symbolism.
3.
The
composition
of
the book is detestable
in that Cardano had
a
habit of
adding
new
thoughts
without
correcting
and/or
eliminating
the
erroneous ones.
4. In
some
instances
Cardano,
in
his
proba
bility
arguments,
creates confusion
by
using
two
entirely
different
methods
without
adequate
distinctions.18
It is unfortunate that the above
in
adequacies
have been obstacles
in
achiev
ing
more
than
a
shallow
understanding
of
De ludo
aleae. We believe that
if
the reader
considers
De
ludo
aleae,
using
the clarifi
cations
provided by
Ore,
he
will
be
im
pressed
by
the
intelligence
and
depth
16. Carl
Boyer,
A
History of
Mathematics
(New
York: John
W?ey
<fe
ons,
1968),
p.
311.
17.
Ore, Cardano,
p.
143.
18.
Ibid.,
pp.
144-45.
30
The
Mathematics
Teacher
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shown
by
Cardano
in
a
virgin
area
of
mathematics.
We
have
attempted
to defend
the
char
acter
and
ability
of
Cardano
not
only
by
examining
the claims of
various
writers
in
the
light
of sixteenth-century moral and
intellectual
values but also
by
referring
to
his
own
arguments.
Clearly,
the
erroneous
statements made
concerning
Cardano's
character
originate
from
ethical
systems
very
different
from
that
in
which Cardano
functioned.
It
is
our
opinion
that
careless
judgments concerning
the
scope
and
quality
of
his works
as
well
as
his
ability
as
a
mathematician
are
in
many
cases
rooted in
this
prejudging
of
Cardano's
character.
Thus
Girolamo
Cardano has
been
maligned
and
castigated unfairly
by
many
modern
writers
who
have
perpet
uated
old
arguments
without
careful
in
vestigation. In sum, we believe that
Cardano,
understood
in
relation
to
his
own
time,
was a
remarkable
individual
and
that he
demonstrated
outstanding
intellect
and
maintained
good
professional
character.
We
invite
the
reader
to
ex
amine
Cardano's
own
work
that is avail
able
in
English,
to
read about
sixteenth
century
life
as
well,
and
then to
judge
Cardano's
character
for
himself.
A
Comment
on
Set Relations
As
is well
known,
the
numbers
of elements of
any
two
finite
sets
A and
B,
their
union,
and
their intersection
re
related
by
(1) n(A KJB)
=
n(A) + n(B)
-
n(A ).
It
is
equally
well
known
that the
lowest
com
mon
multiple
of
any
two
integers
is found
by
dividing
their
product
by
their
highest
common
factor. If
we
write
the
operations
of
finding
the
L.C.M. and the
H.C.F. in
the form
xLy
and
xHy
respectively
{x
and
y
any
two
integers),
we
get
X'V
(2) xLy
=
-J-,
xHy
The
similarity
between
equations
(1)
and
(2)
is
obvious:
the set
operation
U
has
been
replacedby L, byH, and addition and sub
traction
by
multiplication
and
addition.
(If
we
write
(2)
in the
logarithmic
form,
log
(xLy)
?
log
+
log
y
?
log
(xHy),
the
similarity
becomes
even
more
obvious).
Equation
(1)
can
be
extended
to
more
than
two sets: with
the aid
of
the
Boolean
properties
of
the
set
operations,
we can
prove
(3) n(A\JB\JC)
=
n(A)
+
n(B)
+
n(C)
-
n(A
)
-n(BnC)
-n(CnB)
+n(A
HBOC)
Now,
the
operations
L and
H
on
the
natural
numbers have the
same
Boolean
properties
as
the
set
operations
U and
,
if
nstead
of
the
elements of the
sets
we
think
of the
prime
fac
tors
of the
natural
numbers,
and
to
the
empty
set
we
relate
the
number
1.
We then
get
xLyLz
=
(xLy)Lz
_
(xLy)'Z
(xLy)Hz
(by
(2),
applied
to
xLy
and
?)
=
x'y'z
xHy-(xLy)Hz
(by
(2),
applied
to
and
y)
-
x-y-z
~
xtiy{xHz)L(yHz)
(by
distributivity
f
over
L)
_
x-yz-(xHz)H(yHz)
xHyxHz'yHz
(by
(2),
applied
to
xHz
and
and hence
x-yzxHyHz
(4)
xLyLz
=
xHyyHz-zHx
(by
associativity, commutativity
nd
the
idempotent roperty
of
#).
Clearly,
(4)
relates
to
(3)
the
way
(2)
relates
to
(1),
and
(4),
like
(3),
is
capable
of
extension
to
any number of natural numbers.
Akiva
Skidell
Huleh
Valley
Regional
High
School
Kfar
Blum,
Upper
Galilee,
Israel
January
1971
31
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