contraharmonic proportion

contraharmonic proportion

Three positive numbers

x

,

m

,

y

are in

contraharmonic proportion

, if the ratio of the

difference

of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number, i.e. if

m

-

x

y

-

m

=

y

x

.

(1)

The middle number

m

is then called the

contraharmonic mean

(sometimes

antiharmonic mean

) of the first and the last number.

The contraharmonic proportion has very probably been known in the proportion doctrine of the Pythagoreans, since they have in a manner to (1) described the classical Babylonian means:

m

-

x

y

-

m

=

m

m

(

arithmetic mean

⁢

m

)

m

-

x

y

-

m

=

m

y

(

geometric mean

⁢

m

)

m

-

x

y

-

m

=

x

y

(

harmonic mean

⁢

m

)

The contraharmonic mean

m

is between

x

and

y

. Indeed, if we solve it from (1), we get

m

=

x

2

+

y

2

x

+

y

,

(2)

and if we assume that

x

≦

y

, we see that

x

=

x

2

+

x

⁢

y

x

+

y

≦

x

2

+

y

2

x

+

y

≦

x

⁢

y

+

y

2

x

+

y

=

y

.

The contraharmonic mean

c

is the greatest of all the mentioned means,

x

≦

h

≦

g

≦

a

≦

c

≦

y

,

where

a

is the

arithmetic mean

,

g

the

geometric mean

and

h

the

harmonic mean

. It is easy to see that

c

+

h

2

=

a

and

a

⁢

h

=

g

.

Example.

The

integer

5 is the contraharmonic mean of 2 and 6, as well as of 3 and 6, i.e.
2, 5, 6,  are in contraharmonic proportion, similarly are  3, 5, 6:

2

2

+

6

2

2

+

6

=

40

8

=

5

,

3

2

+

6

2

3

+

6

=

45

9

=

5

Note 1.

The graph of (2) is a quadratic cone surface

x

2

+

y

2

-

x

⁢

z

-

y

⁢

z

=

0

, as one may infer of its

level curves

(

x

-

c

2

)

2

+

(

y

-

c

2

)

2

=

c

2

2

which are circles.

Note 2.

Generalising (2) one defines the contraharmonic mean of several positive numbers:

c

⁢

(

x

1

,

…

,

x

n

)

:=

x

1

2

+

…

+

x

n

2

x

1

+

…

+

x

n

There is also a more general

Lehmer mean

:

c

m

⁢

(

x

1

,

…

,

x

n

)

:=

x

1

m

+

1

+

…

+

x

n

m

+

1

x

1

m

+

…

+

x

n

m

References

1

Diderot & d’Alembert

:

Encyclopédie

. Paris (1751–1777).  (Electronic version

http://portail.atilf.fr/encyclopedie/

here).

2

Horst Hischer

: “

http://hischer.de/uds/forsch/preprints/hischer/Preprint98.pdf

Viertausend Jahre Mittelwertbildung”.  —

mathematica didactica

25

(2002). See also

http://www.math.uni-sb.de/PREPRINTS/preprint126.pdf

this.

3

J. Pahikkala

: “On contraharmonic mean and

Pythagorean triples

”.  –

Elemente der Mathematik

65

:2 (2010).

Title

contraharmonic proportion

Canonical

name

ContraharmonicProportion

Date of creation

2015-09-06 19:48:08

Last modified on

2015-09-06 19:48:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Definition

Classification

msc 26E60

Classification

msc 11-00

Classification

msc 01A17

Classification

msc 01A20

Related topic

ProportionEquation

Related topic

Mean3

Related topic

PythagoreanHypotenusesAsContraharmonicMeans

Related topic

HarmonicMean

Related topic

ConstructionOfContraharmonicMeanOfTwoSegments

Related topic

ContrageometricProportion

Defines

contraharmonic mean

Defines

antiharmonic mean

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