The Calculating Machine Logarithm

8/14/2019 The Calculating Machine Logarithm

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M E P M

17-19

P. LYKOUDISC. PAPANAGIOTOU

P. SALMAS; D. SIOUGROS2ND EXPERIMENTAL

LYCEUM OF ATHENS, Greecepanlyk@hotmail .com

history of the much discussed num-ber e cannot start if we do not speak aboutlogarithms in the first place.

In the 16 th 17 th centuries an impor-tant development of the scientific knowl-edge occurred in all fields. The discovery of new lands , the circumnavigation of theEarth by Magellan and the development of sea trade caused the need of map produc-tion ( Gerhard Mercator 1596 ). The intro-duction of Mathematics in Astronomy andPhysics after Copernicus , Galileo and Ke-pler as well as the number of new data ,which needed to be elaborated, requiredcomplicated calculations by the scientists .

There was the need to invent ways thatwould free them from that burden. Since itseasier to add than to multiply, a way of transforming addition into multiplication wasinvented, the logarithm .

John Napier(1550 -1617 ), the8 th Lord of Mer-chiston in Scot-

land , known for hisbooks of religiouscontent, was the firstwho, after havingaccepted the chal-lenge of transform-ing a mathemati-cal action into an

easier one, observed the relation betweenthe terms of a geometrical process andtheir respective exponents that follow ar-ithmetical process .

Napier , using the number 1-10 -7 as a

THE CALCULATING

MACHINE: LOGARITHM

e , .

16 17 . ,

(Gerhard Mercator , 1596 ). ,

,

. .

,

, .

O John Napier (1550 -1617 ), 8 Merchistoun ,

, ,

,

,

. Napier

1-10 -7

:

JOHN NAPIER


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=10 7 (1-10 -7 ) L.

: L= Nap log .

20

Mirifici Logarithmorum Canonis Descriptio .

:

, .

17

, , .

:

:

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basis, suggested that each positive num-ber N can be written as =10 7 (1-10 -7 ) L.

Thus we have the first definition of Ne-perian logarithm : L=Nap log .

In the next 20 years he completed thesuccessive terms of his geometrical proc-ess and finally presented them in his work Mirifici Logarithmorum Canonis De- scriptio .

Note : We have here for the first time the

rate of the sequence , when is toobig, as a base for logarithms .

Money is MathematicsIn the 17 th century an anonymous

trader or usurer noticed a strange behaviourof the interest increase during bank trans-

actions based on compound interest withannual interest divided into equal parts,when number is too big. Lets follow thephenomenon :

The usual banking method of borrowedcapital increase is:

Compound interestIf we deposit K into an account giving

% annual interest and this is repeatedeach year :

At the end of the 1 st year :

At the end of N year : Another usual bank transaction is:

Compound interest times a yearwith annual interest divided into equal parts That is, if we deposit 100 into an accountgiving an interest 5% per year, we will have

11

11

100

1100

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Science and

T echnology

MIRIFICI LOGARITHMORUM CANONIS DESCRIPTIO


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% .

1 :

: :

, 100 5%

:

1 : 105,00 100

5%

(2) 2,5%

1 : 105,06

(4) 1,66%

1 : 105,09

(12) 0,416% . 1 : 105,12

(365) 0,0137 %

1 : 105,19

.

1 1 100

1100

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at the end of each year: At the end of the 1 st year: 105.00 If we deposit 100 into an account giv-

ing an interest 5% per year, twice a year at the end of each semester - we will havean interest of 2.5% :

At the end of the 1 st year: 105.06 ...At the end of each trimester, four times

a year, we will have an interest of 1.66% : At the end of the 1 st year: 105.09 At the end of each month, twelve times

a year, we will have an interest of 0.416% : At the end of the 1 st year: 105.12 At the end of each day, three hundred

sixty five times a year, we will have an inter-est of 0.0137% :

At the end of the 1 st year: 105.19 Suppose we have an interest times a

year. For each period of conversion weconsider as interest rate the annual inter-

est rate divided by , that is % .


Science and

T echnology For =100

1 1

111 2

222 2.25

333 2.37037

444 2.44141

555 2.48832

101010 2.59374

505050 2.69159

100100100 2.70481

100010001000 2.71692

100001000010000 2.71815

100000100000100000 2.71827

100000010000001000000 2.71828

100000001000000010000000 2.71828


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, %.

1 :

: ,

.. / , 2,72 .

e .

e .

e e , ,

.

,

,

:

1100

1

1000000

110000000

1

1

1

1 1 1(1 ) 1 1 ...

2! 3!

1 1 12 ...

2! 3! 4!e

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End of the 1 st year :

Note : The final capital for a very short

period of conversion , e.g. /

, does not exceed the 2.72 of theinitial capital.

We notice that the formula ap-proaches a rate without reaching it and thisrate is number e . In that case, we say that

the sequence in the formula hasnumber e as a limit.

Number e as a limitNumber e is the limit of the sequence ,

as we noted.But since for the high rates of the rate of

will be almost zero,

we have

Consequently :

1100

11000000

110000000

1

1

1

1 1 1(1 ) 1 1 ...

2! 3!

1 1 12 ...

2! 3! 4!e


Science and

T echnology

21 1 ( 1) 1 11 1 ...

2!1 1 2(1 ) (1 )(1 ) 1

1 1 ...2! 3!

21 1 ( 1) 1 11 1 ...

2!

1 1 2(1 ) (1 )(1 ) 11 1 ...

2! 3!


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O Gregorius de Saint-Vincent (1584 -1667 ), , , ,

.

x , , , ... ,

1=1 r ,1+ E2=2(1 r ),1+ E2+ E3=3(1 r )

.

e :

Jacob Bernoulli ( 1654 -1705 )

spira mirabilis ,

x r

xr 2

xr 3

xr

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The quadrature of the hyper-bole

Gregorius de Saint-Vincent (1584 -1667 ) in his effort to square the hyper-

bole notes that, if the x and y coor-dinates of thegraphic functionare changing in ag e o m e t r i c a lprocess , then thearea between theaxis of the coordi-nates and the hy-perbole is chang-ing in an arithme-tic process .

I f an d

If the coordi-

nates x, , , follow a geometri-cal process , the areas

1=1 r ,1+E 2=2(1 r ),1+E 2+E 3=3(1 r )follow an arithmetic

process .So, we conclude that the area between

the axis of the coordinates and the hyper-bole is calculated by the logarithmic func-tion .

When e meets : Logarithmichelix

Jakob Bernoulli (1654 -1705 ) studiedthe logarithmic helix and he named itspira mirabilis because of its mathematicqualities , which make it the most popular

decorative motif apart from the circle .Logarithmic helix is described as a

x

r

xr 2

xr 3

xr


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GREGORIUS DESAINT-VINCENT

IT IS PROVED THAT THE AREAOF THE SPACE =1

: =1


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, , .

(

)

, .

.

Iconography Rick Reed , www.ps.missouri.edu/.../SlideRule02/index.html;www.educ.fc.ul.pt/docentes/opombo/

seminario/neper/biografia.htm;www.nls.uk/scotlandspages/timeline/1614.html;http :// en .wikipedia .org / wiki / File:Gr%C3% A 9goire _ de _ Saint-Vincent _%281584-1667%29.jpg ;http://cwx.prenhall.com/bookbind/pubbooks/thomas_br/chapter1/medialib/custom3/bios/bernoullijakob.htm;http://fridaysunset.net/creation/beauty.html;http://originalbeauty.wordpress. com/2009/06/27/spirals-in-nature.

curve starting from apoint (the pole ) andtwisting in a way thatthe distance betweenits points from the poleincreases in a geomet-rical process , since therotation angle in-creases in an arithmetic

process . Each line pass-ing through the pole in-

cises the helix under the same angle .

Due to the fact that the logarithmic he-lix can be inscribed within harmonic rec-tangles (the ratio of their sides equalsnumber ), it is classified among the har-monic curves .

Logarithmic helix is one of the curves

that can be seen very often in nature .

BibliographyMaor Eli, e: The Story of a Number . Cam-bridge: Cambridge University Press,1994;Eves Haward, Great moments in Mathe- matics - After 1650 . Mathematical Asso-ciation of America,1983;http://en.wikipedia.org/wiki/E_ (mathematical_constant);http://www-history.mcs.st-andrews.ac.uk/history/index.html;http://mathworld.wolfram.com/e.html.


Science and

T echnology

JAKOB BERNOULLI

LOGARITHMICHELIX/SPIRAL GALAXIES

NAUTILUS SHELL WITHLOGARITHMIC SPIRAL E P M

The Calculating Machine Logarithm

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