How log comes into math?

[gimmytang]

I am just wondering when logarithm was first used in math and how people found it.
gim

[Jim Spriggs]

Post by gimmytang
I am just wondering when logarithm was first used in math and how people found it.
gim

The German mathematician Michael Stifel (1486?-1567) in his "Arithmetica
Integra" (date, anyone?) and Chuquet in his "Le Triparty en la science
des numbres" (1484) noted that multiplication/division of terms in a
geometric progression corresponded to addition/subtraction of
exponents. Stifel noted the extension to negative exponents but did not
introduce logarithms as such.

The Scotsman John Napier (1550-1617) noted the correspondence also and
introduced logarithms (c. 1594). His concern was to ease calculation in
spherical trigonometry (essential for long sea voyages) so he invented
logarithms of sines. See his "Mirifici Logarithmorum Canonis
Descriptio" (1614).

Henry Briggs (1561-1631) suggested to Napier in 1615 that the base be
10, and he compiled the first tables of common logarithms from which
were derived the tables in use up until the appearance of the pocket
calculator.

Joost (or Jost?) B\"urgi, a Swiss watchmaker, also noted Stifel's work
and invented logarithms independently of Briggs around 1600, but his
work "Progress Tabulen" was not published until 1620.

Common logarithms are of little interest to mathematicians and have long
since lost their usefulness as a computational aid, but natural
logarithms to base e = 2.718... are important. Euler noted that one may
change base by

log_a(b) = log_c(b)/log_c(a).

Gregory of St Vincent in his "Opus Geometricum" (1647) noted (in modern
notation) that

\int_{t_0)^t dx/x = k log t .

His student Alfons A de Sarasa (1618-67) in "Solutio Problematis a
Mersenno Propositi" (1649) made a similar observation, and maybe Gregory
got the idea from his student.

Newton, around 1665, obtained

log(1 + x) = x - x^2/2 + x^3/3 - ...

where the logarithm is the natural one.

Nicholas Mercator got the same result independently in his
"Logarithmotechnia" (1668).

Let's not be too surprised if someone claims that the Hindus, or
somebody, knew all of this long before the above named were twinkles in
their fathers' eyes.

[m***@usa.net]

Post by Jim Spriggs

Post by gimmytang
I am just wondering when logarithm was first used in math and how people found it.
gim

The German mathematician Michael Stifel (1486?-1567) in his

"Arithmetica

Post by Jim Spriggs
Integra" (date, anyone?) and Chuquet in his "Le Triparty en la

science

Post by Jim Spriggs
des numbres" (1484) noted that multiplication/division of terms in a
geometric progression corresponded to addition/subtraction of
exponents. Stifel noted the extension to negative exponents but did not
introduce logarithms as such.
The Scotsman John Napier (1550-1617) noted the correspondence also and
introduced logarithms (c. 1594). His concern was to ease calculation in
spherical trigonometry (essential for long sea voyages) so he

invented

Post by Jim Spriggs
logarithms of sines. See his "Mirifici Logarithmorum Canonis
Descriptio" (1614).
Henry Briggs (1561-1631) suggested to Napier in 1615 that the base be
10, and he compiled the first tables of common logarithms from which
were derived the tables in use up until the appearance of the pocket
calculator.
Joost (or Jost?) B\"urgi, a Swiss watchmaker, also noted Stifel's work
and invented logarithms independently of Briggs around 1600, but his
work "Progress Tabulen" was not published until 1620.
Common logarithms are of little interest to mathematicians and have long
since lost their usefulness as a computational aid, but natural
logarithms to base e = 2.718... are important. Euler noted that one may
change base by
log_a(b) = log_c(b)/log_c(a).
Gregory of St Vincent in his "Opus Geometricum" (1647) noted (in modern
notation) that
\int_{t_0)^t dx/x = k log t .
His student Alfons A de Sarasa (1618-67) in "Solutio Problematis a
Mersenno Propositi" (1649) made a similar observation, and maybe Gregory
got the idea from his student.
Newton, around 1665, obtained
log(1 + x) = x - x^2/2 + x^3/3 - ...
where the logarithm is the natural one.
Nicholas Mercator got the same result independently in his
"Logarithmotechnia" (1668).
Let's not be too surprised if someone claims that the Hindus, or
somebody, knew all of this long before the above named were twinkles in
their fathers' eyes.

Dear John,
Thank you for your delightful coverage of logarithms, especially your
last remark. That and the fact that there were two independent
"inventors" of logarithms, made me look up John Napier. Let us see
what John Napier did apart from inventing logarithms. I copy below from
a website a description of his inventions:
While better known as a mathematician, John Napier was a busy inventor.
He proposed several military inventions including: burning mirrors that
set enemy ships on fire, special artillery that destroyed everything
within a radius of four miles, bulletproof clothing, a crude version of
a tank, and a submarine-like device. John Napier invented a hydraulic
screw with a revolving axle that lowered water levels in coal pits.
Napier also worked on agricultural innovations to improve crops with
manures and salt.
(http://inventors.about.com/od/nstartinventors/a/John_Napier.htm)
Burning mirrors and a water screw? Are they not attributed to
Archimedes?
Special artillery? Turks had been using artillery for more than a
century! (I know from Indian history that Babar used artillery when he
invaded India circa 1525.) I could not find a mention of the manure and
salt concoction anywhere but I hear fertilizers were in use in Muslim
Spain.
Now let us look at the title of John Napier's book: Mirifici
logarithmorum canonis description. Where does this word "canonis"
come from? (Some folks tell me that the word canon was a direct
"latinization" of the Arabic word "Qanoon". This indicates that
the word canon was in use and was well-understood and so Arabic books
and translations of Arabic books were well received at the time John
Napier wrote his masterpiece. (Some folks also think that
"logarithmourum" also has something to do with Algorithmi or
Al-Khwarizmi. This is all linguistic nonsense, what I want you to
concentrate on is the fact that if Archimedes stole two of John
Napier's inventions, nearly a thousand years before could it be
possible that some Muslim stole from John Napier the invention of
logarithms a few hundred years before? There is also the question of
need. Archimedes' state needed those "inventions", John
Napier's did not! The Muslims needed some kind of computation device
for their tables of trigonometric functions and of positions of stars.
The fact that they made those tables indicates that they had some such
device. What did John Napier need it for? Squat! He "invented"
logarithms to assist others in their computation.

Finally, thank you again. Before I read your post I used to get into
arguments with my Muslim friends saying, "Do not go assigning priority
to Muslims for everything" and that " the logarithm was invented by
John Napier". Now I am in doubt.
Muhammad Zafrullah

[Sylvain Croussette]

***@usa.net wrote:

...

Post by m***@usa.net
Now let us look at the title of John Napier's book: Mirifici
logarithmorum canonis description. Where does this word "canonis"
come from? (Some folks tell me that the word canon was a direct
"latinization" of the Arabic word "Qanoon". This indicates that

Crap, total crap (pardon my english). The latin word 'canon' comes
from the greek kanon meaning a rod. It the same origin as the word
cane in english. The greek kanon was a measuring rod, and by
extension it means a standard, or rule, against which something is
measured. It doesn't come from the arabic. It's possible that canon
and qanoon have the same indo-european root, but one doesn't come from
the other directly.

Post by m***@usa.net
the word canon was in use and was well-understood and so Arabic books
and translations of Arabic books were well received at the time John
Napier wrote his masterpiece. (Some folks also think that
"logarithmourum" also has something to do with Algorithmi or
Al-Khwarizmi. This is all linguistic nonsense, what I want you to

Here http://mathforum.org/library/drmath/view/55579.html it says that
logarithm comes from the greek logos and arithmos, see also
http://www.etymonline.com/index.php?l=l&p=10

Post by m***@usa.net
concentrate on is the fact that if Archimedes stole two of John
Napier's inventions, nearly a thousand years before could it be
possible that some Muslim stole from John Napier the invention of
logarithms a few hundred years before? There is also the question of
need. Archimedes' state needed those "inventions", John
Napier's did not! The Muslims needed some kind of computation device
for their tables of trigonometric functions and of positions of stars.
The fact that they made those tables indicates that they had some such
device. What did John Napier need it for? Squat! He "invented"
logarithms to assist others in their computation.

Logarithms were already known in India well before.

Post by m***@usa.net
Finally, thank you again. Before I read your post I used to get into
arguments with my Muslim friends saying, "Do not go assigning priority
to Muslims for everything" and that " the logarithm was invented by
John Napier". Now I am in doubt.
Muhammad Zafrullah

[Shmuel (Seymour J.) Metz]

In <***@o13g2000cwo.googlegroups.com>, on
05/05/2005

Post by m***@usa.net
Thank you for your delightful coverage of logarithms, especially your
last remark. That and the fact that there were two independent
"inventors" of logarithms, made me look up John Napier.

He just gave the bare bones.

Post by m***@usa.net
Burning mirrors and a water screw? Are they not attributed to
Archimedes?

Yes. Napier may not have know of Archimes's work.

Post by m***@usa.net
There is also the question of need. Archimedes' state needed those
"inventions", John Napier's did not!

Huh? The two states needed the inventions for the same reason; there
was only one Archimedes and only one Napier. The states needed to have
the invention s in more than one place at the same time.

BTW, it's fairly common for the name of a later inventor to get
attached, mostly due to poor communications. An extreme example was
the reluctance of Gauss to publish. I suspect that if you keep
assigning priority to Muslims that you will eventually find something
where it really was a Muslim but not the one that you assigned it to.
IMHO, it makes more sense to just follow the Historical record and see
where it leads; there are much more interesting questions in History
than whether something was first discovered by a Christian, Hindu,
Jew, Muslim or Zoroastrian, such as the question[1] of whether it was
independently discovered by different people at about the same time.

[1] "Steam engine time"

[Lawrence House]

When I was a student all the engineers had slide rules hanging from their belts. Slide rules were a variation of using common logs for calculations by changing multiplication, division, powers and roots to addition and subtraction. Slide rules usually gave accuracy to only 2 or 3 places and you had to supply your own decimal points. If you were a surveyor or navigator and needed more accuracy you had to use tables of common logs. The reason COMMON logs (i.e. base 10 )were used was that they harmonize with our system of numeration and therefore you only needed tables of logs of numbers from 1 to 10 to get the log of any number.
When Werner von Braun came from the German rocket program to this country to head our rocket program he was interviewed regularly by our early T.V. reporters and always had a slide rule hanging, holster style, from his belt. He was asked many questions and he usually answered them by whipping it out and doing the calculations on the spot.
I thought this little bit of history might be of interest to some of you who now have much better calculators, which can do any calculation to 12 places instantly.

[Ed van der Meulen]

thanks

ed

[xmp]

Post by Lawrence House
I thought this little bit of history might be of interest to some of you

who now have >much better calculators, which can do any calculation to 12
places instantly.

i was at a small branch library yesterday, and they still have the log table
books. come to think of it, i've seen them in university libraries as well.
you'd think they'd get rid of them to make space for other books e.g.
Dummies Guide to HP-48.

michael

[Anthony Buckland]

Post by xmp

Post by Lawrence House
I thought this little bit of history might be of interest to some of you

who now have >much better calculators, which can do any calculation to 12
places instantly.
i was at a small branch library yesterday, and they still have the log table
books. come to think of it, i've seen them in university libraries as well.
you'd think they'd get rid of them to make space for other books e.g.
Dummies Guide to HP-48.
michael

When I started University (UBC, BSc in Math, Class of 69) we could take
standard tables into exams, but not electronic calculators (which were
bloody
expensive and only 3- or 4-function anyway). The Statistics department
had banks of those mechanical calculators the size of heavy typewriters
that you wound with a little handle to multiply. _The_ computer had
32K of RAM, but to be fair that was 32K of six-character words.
Program and data editing was done by inserting, replacing or removing
punch cards. Slide rules were rife.

By my fourth year, a lot had changed. Word processing for the multitude
was becoming possible. But the Head of the Classics Department still
had a policy of rejecting, with a zero mark, any essay that appeared to
have been printed by a computer rather than a typewriter.

More random reminiscences of the era that produced Earth's only Moon
landings available on request.

--------------------------------------------------------------

The cabbage rolls, but vodka rocks.

(old Russian folk saying)

[Wayne Brown]

Post by xmp

Post by Lawrence House
I thought this little bit of history might be of interest to some of you

who now have >much better calculators, which can do any calculation to 12
places instantly.
i was at a small branch library yesterday, and they still have the log table
books. come to think of it, i've seen them in university libraries as well.
you'd think they'd get rid of them to make space for other books e.g.
Dummies Guide to HP-48.

They're still useful for learning how logs and exponents work, and
for keeping in practice at doing math without the aid of calculators
or computers. I have books of log tables (and trig tables, etc.) on my
shelves, and still carry and use slide rules (in addition to my HP48GX).

[Dave L. Renfro]

Post by Lawrence House
I thought this little bit of history might be of interest
to some of you who now have much better calculators,
which can do any calculation to 12 places instantly.

Such as 1/2 + 3/4 = 4/6 = 0.666666666667 (exact value, of course).

Dave L. Renfro

[Gib Bogle]

Post by Lawrence House
When I was a student all the engineers had slide rules hanging from their belts. Slide rules were a variation of using common logs for calculations by changing multiplication, division, powers and roots to addition and subtraction. Slide rules usually gave accuracy to only 2 or 3 places and you had to supply your own decimal points. If you were a surveyor or navigator and needed more accuracy you had to use tables of common logs. The reason COMMON logs (i.e. base 10 )were used was that they harmonize with our system of numeration and therefore you only needed tables of logs of numbers from 1 to 10 to get the log of any number.
When Werner von Braun came from the German rocket program to this country to head our rocket program he was interviewed regularly by our early T.V. reporters and always had a slide rule hanging, holster style, from his belt. He was asked many questions and he usually answered them by whipping it out and doing the calculations on the spot.
I thought this little bit of history might be of interest to some of you who now have much better calculators, which can do any calculation to 12 places instantly.

My father, who taught electrical engineering, used to pride himself on
executing calculations with his little slide rule faster than his
students could with their calculators. To a smaller number of
significant figures, of course (although in engineering "significant"
might have a different meaning.)