John Napier

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John Napier

• 1550-1617

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(No Transcript)
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Introduction
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1- A source book in Mathematics, 1200-1800
Edited by D. J. Struik. Harvard pp
11-21. 2- Source book in Mathematics by David
Eugene Smith. Dover. Pp 149-155 3- Classics of
Mathematics Edited by Ronald Calinger. Moore
Publishing Company. pp 253-260 4- The discovery
of logarithms by Napier by H. S. Carslaw, Math.
Gazette, vol. 8 (1915-16) pp 76-84 and
115-119. 5- Napiers logarithmic concept a
reply by Florian Cajori, American Mathematical
Monthly, Vol. 23 (1916) No. 3, pp
71-72. 6- History of the Exponential and
Logaritmic concepts by Florian Cajori, American
Mathematical Monthly, Vol. 20 (1913) Numbers 1
through 7. 7- The Napier Tercentenary
Celebration by Florian Cajori, American
Mathematical Monthly, Vol. 21 (1914) pp
321-323. 8- Doubling the life of the astronomer
from the book Great Moments in Mathematics Before
1650, Dolciani Mathematical Expositions, No. 5,
MAA, pp 182-193. 9- The Great Mathematicians pp
95-96 10 A concise history of Mathematics by D.
J. Struik, Dover pp 95-96 11- John Napier
Logarithm John by Lynne Gladstone-Millar.
Publisher National Museums Of Scotland (January
1, 2004)
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Notes Napier published two books on logarithms
- descriptio and constructio. I have only seen
parts of constructio which has 69 articles. The
first three references seem to have been taken
from a translation by W. R. Macdonald. Although
all three references seem to be almost identical,
I prefer Struiks book because it provides a few
more details. The selection of the articles was
made by W. D. Cairns of Oberlin College according
to D. E. Smith.
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Napier gives the definition of logarithms in
article 26. We will study articles 25 and 26.
Then we will compare Napiers logarithms with
natural logarithms and derive some properties of
Napiers logarithms. We will then study
Napiers construction of logarithmic Tables.
Prof. Carslaws paper is very useful for this
study. Other references include several of
Prof. Cajoris papers and other small articles
from various books. The last reference is
excellent for the study of Napiers life.
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Mirifici logarithmorum cannonis descriptio

• The description of the wonderful cannon of
  logarithms
• Published in Edinburgh in 1614
• The book opens with the following

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Hic liber est minimus, si spectes verba, sed
usum Si spectes, Lector, maximus hic liber
est Disce, scies parvo tantum debere libello Te,
quantum magnis mille voluminibus
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The use of this book is quite large, my dear
friend, No matter how modest it looks, You study
it carefully and find that it gives As much as a
thousand big books
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Mirifici logarithmorum canonis constructio

• The construction of the wonderful cannon of
  logarithms.
• Published posthumously at Edinburgh in 1619

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Definition
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Article 25

• Whence a geometrically moving point approaching
  a fixed one has its velocities proportionate to
  its distances from the fixed one

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• Thus referring to the figure, I say that when the
  geometrically moving point G is at T, its
  velocity is as the distance TS, and when G is at
  1 its velocity is as 1S, and when at 2 its
  velocity is as 2S, and so of the others. Hence,
  whatever be the proportion of the distances TS,
  1S, 2S, 3S, 4S, etc., to each other, that of the
  velocities of G at the points T, 1, 2, 3, 4,
  etc., to one another, will be the same.
• 1 2 3 4
  
• T G G G G
  S

S
T
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• For we observe that a moving point is declared
  more or less swift, according as it is seen to be
  borne over a greater or less space in equal
  times. Hence the ratio of the spaces traversed is
  necessarily the same as that of the velocities.
  But the ratio of the space traversed in equal
  times, T1, 12, 23, 34, etc., is that of the
  distances TS, 1S, 2S, 3S, etc. Hence it follows
  that the ratio to one another of the distances of
  G from S, namely TS, 1S, 2S, 3S etc. is the same
  as that of the velocities of G at the points T,
  1, 2, 3, etc., respectively.

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NotesKeep in mind that Calculus had not been
invented yet. Also, It is important to note that
the exponential notation had not been introduced
yet. Napier establishes that the distances
traveled by a moving point in equal times being
in geometrical progression is equivalent to
saying that the velocity varies as the distance.
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Article 26

• The logarithm of a given sine is that number
  which has increased arithmetically with the same
  velocity throughout as that with which radius
  began to decrease geometrically, and in the same
  time as radius has decreased to the given sine.

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Notes Napier was interested in finding
logarithms of sines of angles. At that time sin
was taken as rsin using the modern notation, r
being the radius of the circle, which acts as the
hypotenuse of the usual right triangle. Thus,
the value of sine depended on the radius that
was chosen. Note that sin90, the largest
sine, would be equal to the radius. Euler started
the modern practice by taking r 1. In this
definition, read radius as a given fixed number
and sine as some other number smaller than or
equal to the radius.
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Let the line TS be the radius, and dS a given
sine in the same line let g move geometrically
from T to d in certain determinate moments of
time. Again, let bi be another line, infinite
towards I, along which from b, let a move
arithmetically with the same velocity as g had at
first, when at T and from the fixed point b in
the direction of i let a advance in just the same
moments of time up to the point c. The number
measuring the line bc is called the logarithm of
the given sine dS.
d
S
T
g
g
c
i
b
a
a
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Notes Referring to the preceding figure Napier
Log dS bc Let us set TS r, dS y and bc x,
Then Napier Log y x
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Using Calculus and the fact that velocity of d
varies as y, we get
ln y -kt C t 0, y r so that C lnr lny
-kt lnr kt ln(r/y) Since initial velocity
is kr, x krt Hence
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Napier log y r ln(r/y)
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Using this it is easy to show that, Nap log uv
Nap log u Nap log v r lnr Nap log (u/v) Nap
log u Nap log v r lnr Also, Nap log (uv/r)
Nap log u Nap log v Nap log (ur/v) Nap log
u Nap log v
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Construction of the table of logarithms
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John Napier proved the following two theorems
using kinematics. He then used these to compute
the logarithms. We will use the modern notation
as used by Prof. Carslaw. Recall that r is the
radius and s is any sine. Thus
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Article 28 Limits of the logarithm

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Article 39 The
difference of the logarithms of two sines
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Napier started out by constructing a geometrical
sequence whose first term is ten million. The
common ratio is chosen to be close to 1 so that
the terms of the sequence remain close to each
other. He used a very clever scheme and
constructed several sequences related to the
first. The last term of the last sequence is
nearly 5 million. This would have been the 6.9
millionth term of the first sequence had it been
continued.
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Recall that Napiers objective was to construct
a table of logarithms of sines. Recall also that
the radius was equal to sine of 90 degrees. The
first term of the sequence is the radius. Thus
the last term is the sine of 30 degrees. Having
completed the table of logarithms of terms of
this sequence Napier shows various ways in which
logarithms of angles between 0 and 30 degrees can
be found.
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I have tried to explain these ideas using
a spreadsheet. The first sequence has 101
terms. All sequences have 10 million for the
first term. The second sequence has the common
ratio equal to the ratio of the first and the
last term of the first sequence. Similarly, the
third sequence has the common ratio equal to the
ratio of the first and the last terms of the
second sequence.
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Since putting these sequences one after
another would not form a geometric sequence,
Napier constructed a table of 21 rows and 69
columns. The first row of this table is a
geometric Sequence with common ratio .99. Each
column is a geometric sequence with common ratio
.9995 He called it table three. Napier
constructed the logarithm of these numbers using
the theorems mentioned before. Note that the
logarithms are very close to the term number had
the first sequence been continued to 6.9 million
terms.
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Napier's tables.xls
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Napier called the table of logarithms the Radical
table. constructio was published posthumously.
Napier had worked on another and better kind
of logarithms but his health was failing. Briggs
worked on these ideas and made improvements and
those are the logarithms that we call the common
logarithms.