Carl Benjamin Boyer - Cardan and the Pascal triangle

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Cardan and the Pascal TriangleAuthor(s): C. B. BoyerSource: The American Mathematical Monthly, Vol. 57, No. 6 (Jun. - Jul., 1950), pp. 387-390Published by: Mathematical Association of America

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1950]

CARDAN AND THE PASCAL

TRIANGLE 387

of

an intersection oint theoremrequires that there be only a finitenumberof

intersection oints

n

the

configuration.hus it is seen that an intersection oint

at

the maximum

distance

from

ny particular point

of the

plane

can

be so iso-

lated. Note that the definition f a

non-trivial ntersection oint theorem lso

requires at least three ines on a pointand threepoints on a line so that we have

at least three

ines

of the

configuration oncurrent t the solated

point

and cut-

ting

the

axis

in

ordinarypoints. With

the configuration

f the theorem

hus es-

tablished

in

a non-desarguesianplane, it is seen that the argumentsof the pre-

ceding

section

apply

as before

o show that

the

theorem s false

n

every

one of

the geometries f class U.

Thus far

we

have only considerednon-trivial onstructible ntersection oint

theorems f

euclidean geometry.

However,

we can show that

possible

theorems

which do

not hold

in

euclidean

geometry

annot hold

in

our

non-desarguesian

planes either.For, considerthe configuration f such a theorem onstructed n a

euclidean plane. Establish the

configuration

n

a non-desarguesian plane by

drawing

n axis

isolating

t

completely

n

the owerhalf-plane.Then,

ifthe theo-

rem

were valid

in

the

non-desarguesiangeometry, t would hold

in euclidean

geometry

lso.

We may

now

state

THEOREM . All non-trivial

onstructiblentersectionointtheoremsrefalse in

each

of

the

geometriesf

class

U.

This yields, by elementarymeans,

the

result of

Moufang (l.c.)

to

the

effect

that all non-trivial onstructible ntersection oint theorems re independent f

the

plane

axioms

of

projective

geometry.

t is

interesting

o note that a

common

geometric

model

(Moulton's

geometry,

or

example)

can be

exhibited

to

show

this

independence

for all non-trivial onstructible ntersection

oint theorems,

and that

each

geometry

f

class

U

will serve as such

a

model.

CARDAN

AND THE PASCAL

TRIANGLE

C. B.

BOYER, Brooklynollege

It is well known that the arithmetic

riangle, n one

or another variant

of

the form

1 1

1

2 1

1

3 3 1

1

4

6

4

1

a

. .

.

. h fau T

d. .

appeared

ong

before ascal

composed

isfamous raitg u

triangle

rithm9tique







388

CARDAN

AND THE

PASCAL

TRIANGLE

[June,

(published

posthumously

n

1665). The device

may

have been known

to

Omar

Khayyam

(c. 1100),

it appeared

in

China

in

1303,

and it

was published

in

Europe

by

Apian

in

1529.

[1] Similar

forms

f the

triangle

were given

before

Pascal

by

numerous

men,

including

Stifel

(1544),

Scheubel

(1545),

Peletier

(1549), Rudolph (1553), Tartaglia (1556), Stevin (1585), Girard (1629), Ought-

red (1631),

and

Briggs

1633).

[2]

However,

it seems

still

to

be widely

believed

[3]

that

the triangular

rray

was

investigated

by Pascal (1654) under

a

new

form,

ubstantially

as follows :

1

1 1

1

1...

1

2 3 4...

1 3

6.*.

1

4--.

1.*-

So significant

id

Cantor, Wieleitner,

nd Tropfke

regard

the

differencen

form

that

theydeclined

to

recognize

any

dependence

of Pascal

upon

the

earlier

n-

stances.

[4]

There

also is

a general

impression

hat

to

Pascal

is due the

first

study

of the

relationships

exhibited

by

the triangle

and

their

application

to

questions

in

the

theory

ofprobability.

[5]

It is

the

purpose

of this

note to

call

attention

to the

work

of one

whose

name

has not

been

associated

with

the

arithmetic

riangle

but

who

anticipated

Pascal with respect

both to the

form

and the study of the triangle.This man, perhaps the greatestmathematician

of

his

day,

was Jerome

Cardan

(1501-1576).

The Ars magna,

which

appeared

in

1545,

contains

no reference

o

the arith-

metic

triangle;

but

in 1570

Cardan

published

his

Opus novum

eproportionibus,

and

in

this work

the Pascal

triangleappears

in both

forms

nd

withvarying

applications.

In

connection

with

the

problem

of the

determination

f

roots

of

numbers,

Cardan

used

the

familiar

earlier

form,

iting

Stifel as the

putative

discoverer.

Here

he gave the

numbers

n

the

triangle

through

n

=

17,

and

he

pointed

out

the relationship,

known

to

Stifel,

equivalent

to

{m8

m

8

m

+

18

(n

)

(n+

1

(

+

1

[6].

Later

on

in

the

Opus

novum

he

question

of

combinations

nd

probabilities

is

taken

up,

and

then

Cardan

gave

the

arithmetic

riangle,

through

n=

11,

in

virtually

he form ater

made famous

by

Pascal. [7] Even

in the

case of

Cardan,

however,

his

form

was

not

entirely

riginal,

for

n

1556

Tartaglia

had published

it

in

a somewhat

similar

square

array,

[8]

and

had

used

it in

determining

he

coefficients

n

theexpansion

of the twelfth

ower

cubo-censo-censo)

f a

binomial

(una quantitadivisiain dueparti). In connectionwiththe newer rrangement f

the

triangle

Cardan

reiterated

the familiar

formula

known

by his

name

and

enunciated by

him

manyyears

before-the

total

numberof different

ombina-







1950]

CARDAN

AND THE PASCAL TRIANGLE

389

tions which can be formed rom

n differentbjects,

taking as manyat a time as

one pleases, is

given by 2

-

1. As an illustration f

the usefulness fhis result,

he determined

he numberof possible

combinationsof 25 different

bjects, the

result being 225_-1=16777215.

Following this, Cardan

gave something

more

important which seems to have been overlookedby historians and which in-

correctlyhas been ascribed to Pascal.

[9] This is the rule of succession,

equiva-

lent to

{n

n-

r+

1(nA

r/ r \r-1

The rule s notgiven symbolically

but is expressed

rather wkwardly s follows:

To obtain the third element

in the row corresponding o n =11, for

example,

subtract one fromeleven, divide by two, and then multiply by eleven. The

result,55, is the number

desired.To obtain the next number, ubtract

two

from

eleven,

divide by three, nd multiplyby

55.

Continuing

n

thismanner,Cardan

obtained

all

of the

elements

n

the

sequence

of

combinations

of

eleven objects.

[10] Had Cardan applied

his rule to the

expansion

of

binomials,

he

would have

anticipated

the binomial

heorem or

positive ntegralpowers.

nstead he

empha-

sized the

connectionbetween the numbers

f the arithmetic

riangle

nd mixed

proportions, .e., progressions

f

higher rder,

nd the

applicability

of

these to

musical

theory.

There can

be

little

doubt

but

that

Cardan,

like

Tartaglia, was

aware that

the

elements

in

the

triangle

are coefficients

n

the

expansions

of

binomials.

That a clear-cut

tatementof Cardan's rule of succession as

applied

to the

binomial

theorem

hould

have

been

delayed

for bout

another

century

s

one

of

the anomalies

in the

development

of

mathematics,

nd that

the

arith-

metic triangle

should be

named

for Pascal

rather

than

for one

of

his

many

anticipators

s

largely

an

accident of

history.

References

1. D.

E.

Smith, istory

fmathematics2 vols.,New

York,

1923-1925),

ol.

2, pp.508-511.

Cf.J.

L. Coolidge,

he story

fthebinomial heorem,

hisMONTHLY, 56, 1949,

47-157.

2. HenriBosmans, nenotehistoriqueur e triangle rithm6tiquee Pascal, Annales e la

Soci6t6 cientifique

e

Bruxelles,

1

(1906-1907),

5-72. Cf.

Smith,

oc.

cit.,

nd

Florian

Cajori,

History

f

mathematics2nd

ed.,New York,

1931),pp. 76,

183,187.

3.

Smith, p.cit.,

nd Coolidge,oc.

cit.

4. Moritz

Cantor,Vorlesungen

ber

Geschichte

er Mathematik

4 vols.,

Leipzig, 1900-

1908),

I, 685-688;

Heinrich

Wieleitner,

eschichte

erMathematik2vols.,

Leipzig, 908-1921),

vol.

II, part1, p.

93; Johannes

ropfke,

eschichteerElementar-Mathematik

2nded.,

7 vols.,

Berlin

ndLeipzig,

921-1924),

I, 34-39.

5. Cajori,

Cantor,

nd Smith, p. cit.;

.

Todhunter,

history

fthemathematical

heory

f

probability

Cambridge

ndLondon, 865),p.

17

f.

6.

Jerome

ardan,Opera

omnia 10vols.,

Lugduni, 663),

V,

529.

7. Ibid., V, 557.

8. Niccolo

Tartaglia,

General

rattato i numeri t

misure part I, Vinegia, 556),

book

1,

fol.

17;

book I,

fols.

9-73.H. G.

Zeuthen,

eschichte

er

Mathematik

m XVI.

undXVII. Jah1r

hundert

German

d.,

Leipzig,1903),

pp.

102,

169,

and Guillaume-ibri,

Histoire es sciences







390

THE

FINITE

FOURIER SERIES AND ELEMENTARY GEOMETRY

[June,

math6matiques

n talie

4 vols., aris, 838-1841),II, 158,mentionhework fTartaglia ut

not

thatof Cardan.

9. Encyclop6die es

sciencesmath6matiques, (1), 1 (1904),pp. 83-84. Tropfke,oc.

cit.,

ascribes ardan's ule f

uccessiono Briggs

n

1633.

10. Operaomnia,

V,

558.

THE FINITE

FOURIER

SERIES

AND

ELEMENTARY

GEOMETRY

I.

J.

SCHOENBERG, University

f

Pennsylvania

Every freshman

knows,

or soon learns,

that the periodic

sequence

xo=l

Xl=-1,

X2=1,

X3=

-1,...

has a general erm

S,

whichmaybe written s

Xn=

(- 1)

.

This is a

very special

case

of

representation

f a

periodic

sequence

by the

so-called finiteFourier

Series. The finite ourier

series

is the

analogue

for periodic sequences

of

the

ordinary

Fourier

series

expansion

of

periodic

functions.

ts only

mention n

our

textbook iterature

s under

the

heading

of

practical

harmonic

analysis,

or

trigonometric

nterpolation, s

found n books

on applied

mathematicsor

numerical

methods; and

yet, the range

of applica-

tions

of the

finite ourier

eriesextends

beyond

this mportant ractical

problem.

It is intimately

related to

the Gaussian sums

and has been

used

for

number

theoreticpurposesby Eisensteinand more recentlyby H. A. Rademacher. The

following

ines

present

he basic

properties

f'the complex)

finite

ourier

series

stressing

ts

geometric

ignificance

nd

followedby a

few applications

to

extre-

mal

problems

of elementarygeometry.

?1.

THE

FINITE

FOURIER SERIES

1. The finite

ourier

eries.We recall

the general

problem

f polynomial

interpolation:

fwe are given

kdistinct

omplex

numberswo,

w1, * *

*,

Wk-1,

and

a second set of k arbitrary

omplex

numbers

o,

1,

. .

.,

Zk-1,

then

there s

one

and only one polynomialP(x)

=o+?jx+

+?k 1Xk-l

satisfying he equa-

tions

P(co,)

=z,

(v=0,

1, *

*,

k-1)

or

(1)

~ ~

~~~2 k-i1L

Zp)o

=

+

1lWp

+

P2WP

+ +

Pk-1WP

(V

=

0

12

.

.

.

2

k

-

1). [1]

Indeed,

the

system

of

inear equations

(1)

in the unknowns

Rp

as a

non-vanish-

ing,

Vandermonde)

determinantnd has

therefore

unique

solution.

We

obtain

the

finiteFourier

series

(abbreviated in

the sequel to f.F.S.)

if we

choose the

numbers

op

o be

the

kth

roots

of unity; so for

the remainder

of

this

paper

we

shall

assume that

(2)

w,

=

e2liPk

(v= 0, 1, k-1).

This equation

defines

op

or ll integral

values of

v,

a fact

occasionally

used

later.





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