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You go so much informative material. Reall a great treasure, and
thanks.
--- In sliderule@yahoogroups.com, "Bill Robinson" <wrobinson62@c...>
wrote:
>
> Ken:
>
>
>
> The Gudermanian is important as its value allows the calculation of
the
> hyperbolic functions from the circular functions. This was found
to be a
> very convenient way to develop tables of hyperbolic functions. In
1832
> Christof Gudermann published extensive tables of Log Sinh u and Log
Cosh u
> from values calculated from the Gudermannian function. The
connection
> between the hyperbolic and gudermannian functions are as follows:
>
> sinh u = tan(gd u); cosh u = sec(gd u); and tanh u = sin (gd
u). Where
> the gudermannian of u = gd u = 2 arctan e^u - p/2
>
>
>
> The above is just a short description. I have a lot more about the
gd if you
> need it. Let me know if you still have questions.
>
> Of course today with our calculators and computers we do not need
theses
> tables. But back in the early 1800's Gudermann's tables were
important
> enough in those years to earn him a place in history with the
> "Gudermannian". Some more information follows below.
>
>
>
>
>
> Excerpts from JOS Vol. 14, No. 1 2005 in reference to the
Gudermannian
>
>
>
> Listing of Slide Rules with Hyperbolic Functions
>
> By William K. Robinson
>
> A Short History of Hyperbolic Functions
>
> This short history has been adopted mainly from Hyperbolic Functions
> (Smithsonian Mathematical Tables) prepared by George F. Becker and
C. E. Van
> Orstand, 1909 (Second reprint 1920). I reviewed a few other
references on
> the history of mathematics to confirm the important dates and people
> involved.
>
> Hyperbolic functions were not introduced until around the
1760's.
> However, it was some two hundred years earlier that the first and
one of the
> most important applications of the functions now known as
hyperbolic was
> made by Gerhard Kremer, 1512-1594, the Flemish geographer who was
better
> known by his Latin name, Gerhardus Mercator. In 1569 he issued his
> Mercator's Projection map on which the loxodrome was a straight
line. His
> projection resulted in the making of a map in which a straight line
(the
> loxodrome) always made an equal angle with every meridian. This was
a
> significant breakthrough in navigation. Its importance is evidenced
by the
> fact that today all deep-sea navigation charts of the world have as
their
> basis this projection. Mercator published his map without
explanation and it
> was left to others following him to discover the formulas he used.
These
> were later found to be: l = gd (m/a), and (m/a) = ln tan [(p/4) +
(l/2)],
> where l is the latitude, m is the projection point in latitude l,
and a is
> the radius of the Earth. The term gd is called the gudermannian
(after
> Chistoph Gudermann, 1798-1851). Of interest is the relationship of
gd to
> hyperbolic functions. This was subsequently found to be as
follows: tan gd
> x = sinh x; sec gd x = cosh x; and sin gd x = tanh x.
>
>
>
> Vincenzo Riccati (1707-1775) is noted as the actual
inventor of
> hyperbolic trigonometry. In his publications in the 1760's he
introduced the
> use of hyperbolic functions to obtain the roots of certain types of
> equations, particularly cubic equations. He adopted the notation
Sh.f and
> Ch.f for the hyperbolic functions, and Sc.f and Cc.f for the
circular ones.
>
>
> Soon after Daviet de Foncenex showed how to interchange
circular
> and hyperbolic functions by the use of i (v-1). (It was Euler
(1748) who
> introduced the symbol i for v-1. Also, we cannot forget De Moivre's
> well-known contributions that were generalized by Euler to: (cos x
+ i sin
> x)n = cos nx + i sin nx). From Euler and these earlier efforts
emerged the
> now familiar circular and hyperbolic equations: sin a = - i sinh
ia , and
> cos a = cosh ia. These are matched by the converse equations: sinh
b = - i
> sin ib, and cosh b = cos ib. Using these above equations
substitutions could
> be readily made between the circular and hyperbolic functions.
>
> The first systematic development of Hyperbolic Functions
was by
> Johann Heinrich Lambert (1728-1777). He adopted the notation we
use today;
> sinh u, cosh u, etc., and introduced the transcendent angle (later
renamed
> the gudermannian) using it in computation and in the construction
of tables.
> He is credited with popularizing the new hyperbolic trigonometry
that modern
> science finds so useful. It has been said that Lambert did for
hyperbolic
> functions what Euler had done for circular functions. In 1830,
Gudermann
> published an important paper followed by extended tables. Then
Cayley, in
> 1862, in recognition of Gudermann's contributions, proposed the name
> gudermannian for the angle that Lambert called transcendent. The
name,
> gudermannian, remains today.
>
> Slide Rules with Hyperbolic Scales
>
> As the expansion of applications accelerated, it seemed only a
matter of
> time until the slide rule emerged as an aid for the Electrical
Engineer.
> The history of slide rules with hyperbolic scales begins with a
Patent
> application on May 12, 1921 by Albert F. Puchstein. The title
was ""Device
> For Making Vector Calculations", and included layouts of hyperbolic
scales.
> In the patent application Puchstein says; "::.my device is of such
a nature
> that calculations can be readily made as to :.. hyperbolic sines,
cosines,
> tangents, etcetera, of vectors". It is interesting to note that a
period of
> nine years went by from inception to introduction of the K&E 4093-3
in the
> market. For although the patent was approved three years later on
March 25,
> 1924 (No. 1,487,805), it was not listed for sale in the K&E
catalogs until
> 1930, when the manual was published. The rule's price was $16.00.
The
> 4093-3s version, with a better leather case, was $16.85.
>
> Next in line it appears was the Hemmi Model 153 (1933)
that had
> Gudermannian (spelled Gudermanian by Hemmi) scales for obtaining
Hyperbolic
> Functions. Its 1934 manual describes this model as the Electrical
Engineer's
> Universal Duplex Slide Rule With Patent Vector Scale, Gudermanian
Scale and
> Log Log Scales. This is a very usable rule and is one of the most
uniquely
> designed of all the hyperbolic slide rules.
>
> Most of the rules have designated hyperbolic Sinh and Tanh
scales.
> (A few exceptions are the Hemmi 153, and similar rules from other
makers,
> that have Gudermannian scales for obtaining Hyperbolic Functions).
Usually
> the scales are shown as Sh1, Sh2 and Th. A few have a hyperbolic
Cosh scale
> shown as Ch. Almost always there are two hyperbolic Sh scales to
give a
> wider range for accuracy. The ranges of the Hyperbolic Scales
usually are:
> Sh1 from 0.1 to 0.882; Sh2 0.882 to 3.0; and Th from 0.1 to 3.0.
Only a few
> rules include a Cosh (Ch) scale in the layout. When the Cosh scale
is
> included on the rule the range of values is usually Ch from 0.1 to
3.0. For
> those rules that do not have a Ch scale its value is calculated by
use of
> the formula Ch = Sh / Th.
>
> With the slide reversed on the K&E 4083-3 the scales
layout becomes
> almost ideal, and the accuracy and ability to handle all
calculations
> involving hyperbolic functions is greatly enhanced. At almost a tie
for ease
> of use is the Hemmi No. 153. This rule, and three others on the
list are
> gudermannian in design. With the slide reversed on this slide rule
you can
> read off the Sinh and Tanh values on the T and Q scales with one
setting on
> the Gq scale. Then sliding the hairline to the Sinh value on the Q
scale you
> can read the Cosh value on the Q' scale. Other rules that work well
for
> hyperbolic functions with the slide reversed are as follows: Aristo
0971,
> Aristo 0972, Dietzgen 1735, Ding Feng 5471, Flying Fish 1004, all
Pickett
> Model 4's, and SIDA Models 1, 1083 and 6201.
>
>
> Best regards, Bill Robinson (Email: wrobinson62@c...
>
>
>
>
>
>
> [Non-text portions of this message have been removed]
>
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