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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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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







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







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







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







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







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







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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James K. Bidwell y Bernard K. Lange - Girolamo Cardano. A defense of his character