Re: Computing hour lines for horizontal sundials

[Michael Ossipoff]

As others have pointed out, you don’t need the logarithms. Tables of
logarithms, trig-functions, & logs of trig-functions aren’t needed now that
we have scientific-calculators, computers, spreadsheets, etc. Just use the
trig-functions directly, as the others have said.

.

You made a good choice when you chose the Horizontal-Dial. There are good
reasons why it’s the most popular stationary dial.

.

For one thing, it’s the easiest one to build, & also the easiest to set up.

.

For another thing, the marking of its hour-lines is easily explained

.

Additionally, it can be read from any direction (though you have to stand
fairly near to it), & it tells time whenever the Sun is up (unless it’s
shaded by something).

.

About the explanation of the marking of the Horizontal-Dial’s hour-lines:

.

Start with a Disk-Equatorial. It’s a circular disk with equally-spaced hour
lines radially marked around the gnomon that passes through the disk’s
center, perpendicularly to the disk. It’s mounted so that the disk is
parallel to the equator. It has a stick-gnomon going through it, through
its center, perpendicular to the disk, & therefore parallel to the Earth’s
axis.

.

The relation between the gnomon-stick’s length down from the lower face of
the disk, & the disk’s diameter can be chosen so that when the device is
laid on the ground, resting on the disk-edge & on the bottom-end of the
long gnomon-stick, the gnomon will be parallel to the Earth’s axis .
Disk-Equatorials have been mounted directly on the ground in that form. I
once read that dials of that type were the earliest known sundials.

.

So, to make a Horizontal-Dial from an Equatorial-Disk Dial of that type:

.

Extend, project, the hour-lines to the ground.

.

The projected lines will intersect the ground along an east-west line.

.

To each of those intersections, draw a line from the point where the
bottom-end of the gnomon-stick touches the ground.

.

Those are the hour-lines of the Horizontal-Dial.

.

The 6:00 line wouldn’t intersect the ground, because the line would be
horizontal, but it’s evident that, the closer the time is to 6:00, the more
closely the direction of the Horizontal-Dial’s hour-line approaches
perpendicular to the noon hour-line. So just make the 6:00 hour-line
perpendicular to the hour-line.  (That’s neatly automatically achieved by
the formula.)

.

Of course, on the summer side of the equinoxes, the day will start before
6:00 a.m., & end after 6:00 p.m.  For those times’ hour-lines, just (for
example) extend the 7:00 a.m. hour-line across the dial-plate, to make the
7:00 p.m. line.   …doing the same for the other p.m. times that have
sunshine.

.

When you read the definitions of the sine & the tangent, it will be obvious
that the formula is just a mathematical expression of the above-described
method for constructing the hour-lines of the Horizontal Dial.

On Tue, Aug 9, 2022 at 1:50 AM Bryan Mumford  wrote:

> I’m working from Albert Waugh’s book “Sun dials, Their Theory and
> Construction”. On page 45 he presents a method for computing hour lines. I
> lack significant math skills, but I know how to work Excel. I don’t
> understand how he is calculating these values.
>
> He says, for example, that “log tan t” of 7°30’ is 9.11943.
>
> In my simple-minded way I asked Excel to show me log(tan(7)) and got a
> very different value.
> I tried converting 7°30’ to radians and that didn’t get any closer.
>
> How can I calculate "log tan t" or "log sin latitude” with Excel to get
> the values he shows?
>
> I anticipate further problems with the last two columns, but you have to
> start somewhere….
>
> - Bryan
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
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Re: Computing hour lines for horizontal sundials

[Frank King]

Dear Steve,

R. Hooijenga is right.  Adding 10 was a common way to proceed.

We are talking about log-to-base-10 which I imagine is rarely used these
days but when it was in common use there were all kinds of tricks. I
suspect these were different in North America from in Europe

Consider the logs of 2, 20, 200 etc, we have:

 0.3010, 1.3010, 2.3010 etc

Easy. Now start going down, 0.2, 0.02, 0.002 etc.  Your way sounds like a
nightmare to me.  The way I was taught was (almost) to write:

 -1+0.3010, -2+0.3010, -3+0.3010 etc

except we actually wrote 'one-bar...' instead of '-1+0...'.

Roger Bailey clearly thinks this is some kind of archaic religion but if
you are going to understand an ancient god like Waugh you had better study
his liturgy!!

Very best wishes

Frank


On Tue, Aug 9, 2022, 10:43 Steve Lelievre 
wrote:

Ooof!
>
> Did the method of adjusting all the logs by +10 really make the task
> easier?
>
> Merely negating the log seems better to me or simply learning to do
> arithmetic on negatives.
>
> Steve
>
---
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Re: Computing hour lines for horizontal sundials

[Roger Bailey]

Hello Bryan,
Cut to the chase. Use the trig equations directly. Solve with a simple
scientific calculator. No worries about logs or radians.

My background is similar to most of the others who provided good advice on
old fashion methods.  Over the last 50 years I have junked log tables,
sight reduction tables, my log log slide rules and most of my old
programmable calculators.

Welcome to the arcane abstruse world of sundial design.

Roger Bailey, Peng
Walking Shadow Designs
 Sidney by the Sea, BC

On Mon, Aug 8, 2022 at 10:50 PM Bryan Mumford  wrote:

> I’m working from Albert Waugh’s book “Sun dials, Their Theory and
> Construction”. On page 45 he presents a method for computing hour lines. I
> lack significant math skills, but I know how to work Excel. I don’t
> understand how he is calculating these values.
>
> He says, for example, that “log tan t” of 7°30’ is 9.11943.
>
> In my simple-minded way I asked Excel to show me log(tan(7)) and got a
> very different value.
> I tried converting 7°30’ to radians and that didn’t get any closer.
>
> How can I calculate "log tan t" or "log sin latitude” with Excel to get
> the values he shows?
>
> I anticipate further problems with the last two columns, but you have to
> start somewhere….
>
> - Bryan
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
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Re: Computing hour lines for horizontal sundials

[Steve Lelievre]

Hi,

Actually, I understood that the tables come with 10 already added.

I saw the advantage of having tables of logs of trig values but I have 
struggled to understand the advantage of adding 10 to the values in the 
table. When I compare that to using the logs directly, it seems more 
complicated in that one has to subtract some multiple of ten from the 
sum of the values taken from the table (1 x 10 when multiplying two 
values, 2 x 10 when multiplying three values, etc.) My thinking was that 
if the aim was to simply to avoid using negatives, the calculator-person 
could simply leave off the minus sign.


But now, from looking again at the examples in your diagram, I think I 
understand: it seems that one does not carry digits over into the tens 
column of the summation. And dropping the tens column has the same 
effect as subtracting the extra multiples of 10.


So, for example, the method involves something along the lines of 9.1 + 
9.1 + 9.1 = 27.3 but don't write '2' in the tens column  = 7.3.


It's nice to encounter so many strange things on the Sundial List.

Steve




On 2022-08-09 1:12 p.m., R. Hooijenga wrote:


Hi Steve, all,

Yes, the ’10-trick’ was so common because it made things very easy – 
well, comparatively speaking.


But I see I have not been entirely clear: I forgot to mention the big 
trick, because to me it is so obvious – the user doesn’t have to do 
any adjusting, because the tables list everything ready-made.


For instance, the sines table would not tabulate sines, nor would it 
tabulate the log of sines: it would tabulate the ten plus the log of 
the sine.


All the computer (the person doing the computing!) had to do was look 
up the angle – say, 31° 25’ from the example – and get the number 
9.71705 /directly from the SIN table/. Likewise, 45° 05’ will give you 
9.84885 in the COS table.


The addition of ‘minus ten’ in the example below was just to make it 
clear to me, the student, what was happening. In actual practice it 
was never written down.


And going the other way, still in the example below, you could just 
search for ‘9.88400’ in the log-sin table and find the corresponding 
angle 49°57’ 36”. (Unfortunately, there is a printer’s error in the 
example here: the number should really be 9.88400 , not 0.88400.)


Of course, interpolation was most always required; there were handy 
small lists for that in the margins of the table pages.


/A sight reduction form was a marvel of efficiency/. Just take your 
sextant-read altitudes, determine all necessary corrections (you must 
do all that even today), and enter all on the form.


Then, just proceed line by line: adding, sometimes subtracting, and 
looking up in tables; and you end up with a star fix.


Later, we got the HO-249 (and similar) publications, reducing the work 
even further. I bet that old first mate could work out a fix just as 
fast as anyone can today on an iPad.


And if we dropped our HO-249, the worst that could happen was that we 
cracked the spine (not that it ever happened); compare that to the 
drama that a falling iPad might engender!


Rudolf

*Van:* sundial  *Namens *Steve Lelievre
*Verzonden:* dinsdag 9 augustus 2022 17:43
*Aan:* sundial@uni-koeln.de
*Onderwerp:* Re: Computing hour lines for horizontal sundials

Ooof!

Did the method of adjusting all the logs by +10 really make the task 
easier?


Merely negating the log seems better to me or simply learning to 
do arithmetic on negatives.


Steve
---
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Re: Computing hour lines for horizontal sundials

[Bryan Mumford]

Thanks to all who offered their help!

I confess to knowing nothing about how logs are used, but it would be a 
substantial sidetrack from the task at hand to educate myself on this now.

The Excel expression by Peter Meyer is a huge help. It recreates the values 
reached by Waugh in my spreadsheet and should allow me to produce local dials. 
I NEVER would have gotten there on my own!

=DEGREES(ATAN(TAN(RADIANS(B3))*SIN(RADIANS(latitude

- Bryan



> On Aug 9, 2022, at 4:08 AM, Peter Mayer  wrote:
> 
> Hi Brian,
> 
>  In my experience, using Excel is an excellent way to explore sundial 
> formulas.
> 
> I wouldn't use logs. Instead use the equation above it: Tan D (hour angle) = 
> (tan t (time)*sine phi (latitude)). So I think you need 
> =DEGREES(ATAN(TAN(RADIANS(B3))*SIN(RADIANS(35, i.e. at the end, you need 
> to convert the answer in radians back to degrees. If I use 35 degrees as 
> latitude, I get 8.73785 degrees = 8 44 which is what Waugh lists on p. 219.
> 
> 
> best wishes,
> 
> Peter
> 
> On 9/08/2022 15:19:54, Bryan Mumford wrote:
>> CAUTION: External email. Only click on links or open attachments from 
>> trusted senders.
>> 
>> I’m working from Albert Waugh’s book “Sun dials, Their Theory and 
>> Construction”. On page 45 he presents a method for computing hour lines. I 
>> lack significant math skills, but I know how to work Excel. I don’t 
>> understand how he is calculating these values.
>> 
>> He says, for example, that “log tan t” of 7°30’ is 9.11943.
>> 
>> In my simple-minded way I asked Excel to show me log(tan(7)) and got a very 
>> different value.
>> I tried converting 7°30’ to radians and that didn’t get any closer.
>> 
>> How can I calculate "log tan t" or "log sin latitude” with Excel to get the 
>> values he shows?
>> 
>> I anticipate further problems with the last two columns, but you have to 
>> start somewhere….
>> 
>> - Bryan
>> 
>> 
>> 
>> ---
>> https://lists.uni-koeln.de/mailman/listinfo/sundial 
>> 
>> 
>> 
> -- 
> ---
> Peter Mayer
> Department of Politics & International Relations (POLIR)
> School of Social Sciences
> http://www.arts.adelaide.edu.au/polis/ 
> 
> The University of Adelaide, AUSTRALIA 5005
> Ph : +61 8 8313 5609
> Fax : +61 8 8313 3443
> e-mail: peter.ma...@adelaide.edu.au 
> CRICOS Provider Number 00123M
> ---
> 
> This email message is intended only for the addressee(s) 
> and contains information that may be confidential 
> and/or copyright. If you are not the intended recipient 
> please notify the sender by reply email 
> and immediately delete this email. 
> Use, disclosure or reproduction of this email by anyone 
> other than the intended recipient(s) is strictly prohibited.
> No representation is made that this email or any attachment
> are free of viruses. Virus scanning is recommended and is the
> responsibility of the recipient.
> --
> https://www.adelaide.edu.au/study/ 
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RE: Computing hour lines for horizontal sundials

[R. Hooijenga via sundial]

Diese Nachricht wurde eingewickelt um DMARC-kompatibel zu sein. Die
eigentliche Nachricht steht dadurch in einem Anhang.

This message was wrapped to be DMARC compliant. The actual message
text is therefore in an attachment.--- Begin Message ---
Hi Steve, all,

 

Yes, the ’10-trick’ was so common because it made things very easy – well, 
comparatively speaking.

But I see I have not been entirely clear: I forgot to mention the big trick, 
because to me it is so obvious – the user doesn’t have to do any adjusting, 
because the tables list everything ready-made.

For instance, the sines table would not tabulate sines, nor would it tabulate 
the log of sines: it would tabulate the ten plus the log of the sine.

All the computer (the person doing the computing!) had to do was look up the 
angle – say, 31° 25’ from the example – and get the number 9.71705 directly 
from the SIN table. Likewise, 45° 05’ will give you 9.84885 in the COS table.

The addition of ‘minus ten’ in the example below was just to make it clear to 
me, the student, what was happening. In actual practice it was never written 
down.

 

And going the other way, still in the example below, you could just search for 
‘9.88400’ in the log-sin table and find the corresponding angle 49° 57’ 36”. 
(Unfortunately, there is a printer’s error in the example here: the number 
should really be 9.88400 , not 0.88400.)

Of course, interpolation was most always required; there were handy small lists 
for that in the margins of the table pages.

 

A sight reduction form was a marvel of efficiency. Just take your sextant-read 
altitudes, determine all necessary corrections (you must do all that even 
today), and enter all on the form.

Then, just proceed line by line: adding, sometimes subtracting, and looking up 
in tables; and you end up with a star fix.

 

Later, we got the HO-249 (and similar) publications, reducing the work even 
further. I bet that old first mate could work out a fix just as fast as anyone 
can today on an iPad. 

And if we dropped our HO-249, the worst that could happen was that we cracked 
the spine (not that it ever happened); compare that to the drama that a falling 
iPad might engender!

 

Rudolf

 

Van: sundial  Namens Steve Lelievre
Verzonden: dinsdag 9 augustus 2022 17:43
Aan: sundial@uni-koeln.de
Onderwerp: Re: Computing hour lines for horizontal sundials

 

Ooof!

 

Did the method of adjusting all the logs by +10 really make the task easier?

 

Merely negating the log seems better to me or simply learning to do 
arithmetic on negatives.

 

Steve

 



 

--- End Message ---
---
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Re: Computing hour lines for horizontal sundials (Bob Kellogg)

[Robert Kellogg]

Bryan, Steve

In Engineering it is common to use 10*log10(x)  for example if x=3 
10*log10(3) = 4.77.  This has the advantage that 10*log10(10) = 10. Many 
times these are described as units of decibels (dB).  For example 3 
watts is 4.77 dBW (decibel watts).   Or 10 milliwatts is 10 dBm.   When 
studying the laws of RF propagation or the strength of gravitational 
force where in each case quantities are squared, dB = 10*log10(x^2) = 
20*log10(x).  So these precursor of multiplying by 10 or 20 becomes very 
useful.




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Re: Computing hour lines for horizontal sundials

[Steve Lelievre]

Ooof!

Did the method of adjusting all the logs by +10 really make the task easier?

Merely negating the log seems better to me or simply learning to do 
arithmetic on negatives.


Steve


On 2022-08-09 8:11 a.m., R. Hooijenga via sundial wrote:



Frans' answer is much to the point here.

When I started at sea, star fixes were computed on a sight reduction 
form. Without the benefit of a calculator, it would be folly to 
attempt this without logarithms. (At the time, I did have a brand-new 
Sinclair Scientific, but the first mate took a dim view of this 
new-fangled contraption.)


To avoid the negatives, 10 was added, only to be dismissed after the 
addition or subtraction.


The example below is from W.L. Kennon: /Astronomy/, which I kept from 
freshman year. It shows the use of logarithms and of the ‘10’.


---
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Re: Computing hour lines for horizontal sundials

[Bill Gottesman]

I'm pretty sure that for excel, you would need to enter

=LOG(TAN(RADIANS(7)))

because the trig functions in excel work on radians rather than degrees.
-Bill

On Tue, Aug 9, 2022 at 1:50 AM Bryan Mumford  wrote:

> I’m working from Albert Waugh’s book “Sun dials, Their Theory and
> Construction”. On page 45 he presents a method for computing hour lines. I
> lack significant math skills, but I know how to work Excel. I don’t
> understand how he is calculating these values.
>
> He says, for example, that “log tan t” of 7°30’ is 9.11943.
>
> In my simple-minded way I asked Excel to show me log(tan(7)) and got a
> very different value.
> I tried converting 7°30’ to radians and that didn’t get any closer.
>
> How can I calculate "log tan t" or "log sin latitude” with Excel to get
> the values he shows?
>
> I anticipate further problems with the last two columns, but you have to
> start somewhere….
>
> - Bryan
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
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Re: Computing hour lines for horizontal sundials

[Peter Mayer]

Hi Brian,

 In my experience, using Excel is an excellent way to explore sundial 
formulas.


I wouldn't use logs. Instead use the equation above it: Tan D (hour 
angle) = (tan t (time)*sine phi (latitude)). So I think you need 
=DEGREES(ATAN(TAN(RADIANS(B3))*SIN(RADIANS(35, i.e. at the end, you 
need to convert the answer in radians back to degrees. If I use 35 
degrees as latitude, I get 8.73785 degrees = 8 44 which is what Waugh 
lists on p. 219.



best wishes,

Peter

On 9/08/2022 15:19:54, Bryan Mumford wrote:
CAUTION: External email. Only click on links or open attachments from 
trusted senders.


I’m working from Albert Waugh’s book “Sun dials, Their Theory and 
Construction”. On page 45 he presents a method for computing hour 
lines. I lack significant math skills, but I know how to work Excel. I 
don’t understand how he is calculating these values.


He says, for example, that “log tan t” of 7°30’ is 9.11943.

In my simple-minded way I asked Excel to show me log(tan(7)) and got a 
very different value.

I tried converting 7°30’ to radians and that didn’t get any closer.

How can I calculate "log tan t" or "log sin latitude” with Excel to 
get the values he shows?


I anticipate further problems with the last two columns, but you have 
to start somewhere….


- Bryan



---
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--
---
Peter Mayer
Department of Politics & International Relations (POLIR)
School of Social Sciences
http://www.arts.adelaide.edu.au/polis/
The University of Adelaide, AUSTRALIA 5005
Ph : +61 8 8313 5609
Fax : +61 8 8313 3443
e-mail:peter.ma...@adelaide.edu.au
CRICOS Provider Number 00123M
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Use, disclosure or reproduction of this email by anyone
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Re: Computing hour lines for horizontal sundials

[Hank de Wit]

Hi Bryan,

The "9" in "9.11943" can't be right for any reasonable "log" of "tan", 
so it's likely a misprint. The "log" than Waugh would likely be using is 
the common log, LOG10, rather than the natural log we mostly use these days.


And indeed LOG10(TAN(7.5)) is −0.8805708975, which is "-1 + 0.1194291025".

In Logarithm books (for those of us who remember using them), when you 
looked up the logarithm of a number you had to first express the number 
in scientific notation as the lookup tables only had numbers in a 
certain range. So in this case, the logarithm in base 10 of 7.5 degrees 
is 0.1316524976, but you had to first express that as 1.316524976 x 
10^(-1), then lookup the logarithm (base 10) of 1.316524976, which is 
0.1194291025. The logarithm of the exponent is -1. However rather than 
waste time subtracting 1 from 0.1194291025 immediately, the exponents 
(whole numbers) and fractions were treated separately in further 
calculations and only combined at the end. Hope that makes sense.


I suspect Waugh was just using logarithms as a means for making 
calculations by paper easier. They would not be inherently part of the 
equations he was trying to solve. He was really interested in the TAN, 
and finding the LOG10, was just so he could add intermediate numbers 
rather than multiply them.


Regards Hank

On 9/8/22 3:19 pm, Bryan Mumford wrote:

I’m working from Albert Waugh’s book “Sun dials, Their Theory and 
Construction”. On page 45 he presents a method for computing hour lines. I lack 
significant math skills, but I know how to work Excel. I don’t understand how 
he is calculating these values.

He says, for example, that “log tan t” of 7°30’ is 9.11943.

In my simple-minded way I asked Excel to show me log(tan(7)) and got a very 
different value.
I tried converting 7°30’ to radians and that didn’t get any closer.

How can I calculate "log tan t" or "log sin latitude” with Excel to get the 
values he shows?

I anticipate further problems with the last two columns, but you have to start 
somewhere….

- Bryan



---
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---
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Re: Computing hour lines for horizontal sundials

[Jan Bielawski]

The presence of the log seems weird in this context. Perhaps a photo of the
relevant page would help.
(I don't have the book handy.)

--
Jan

On Mon, Aug 8, 2022 at 10:50 PM Bryan Mumford  wrote:

> I’m working from Albert Waugh’s book “Sun dials, Their Theory and
> Construction”. On page 45 he presents a method for computing hour lines. I
> lack significant math skills, but I know how to work Excel. I don’t
> understand how he is calculating these values.
>
> He says, for example, that “log tan t” of 7°30’ is 9.11943.
>
> In my simple-minded way I asked Excel to show me log(tan(7)) and got a
> very different value.
> I tried converting 7°30’ to radians and that didn’t get any closer.
>
> How can I calculate "log tan t" or "log sin latitude” with Excel to get
> the values he shows?
>
> I anticipate further problems with the last two columns, but you have to
> start somewhere….
>
> - Bryan
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
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Computing hour lines for horizontal sundials

[Bryan Mumford]

I’m working from Albert Waugh’s book “Sun dials, Their Theory and 
Construction”. On page 45 he presents a method for computing hour lines. I lack 
significant math skills, but I know how to work Excel. I don’t understand how 
he is calculating these values.

He says, for example, that “log tan t” of 7°30’ is 9.11943.

In my simple-minded way I asked Excel to show me log(tan(7)) and got a very 
different value.
I tried converting 7°30’ to radians and that didn’t get any closer.

How can I calculate "log tan t" or "log sin latitude” with Excel to get the 
values he shows?

I anticipate further problems with the last two columns, but you have to start 
somewhere….

- Bryan



---
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