Conference rusure::math

Put a stick in the ground.
At different hours of the day, mark with the number of stones equal to
the number of the hour where the shadow of the stick is at that hour.
It is best of the surface is dead level, by 2-dimensional bubble level,
and the stick is set straight with a plumb line, and in a concrete
base.  Let me know if it works all year.
Tom

What you're probably interested in is called the 'equation of time'
and is documented in publications such as the Nautical Almanac.  This relates
how fast or slow local solar time (seen by the sundial) is with respect
to clock time.  You can get a close approximation by simply solving
Kepler's equation using the eccentricity of the earth's orbit, the
inclination of the earth's axis, and the orbital and rotational rates
of the earth...  its pretty simple geometry.

I once programmed this out of curiosity, and may be able to find the
routines somewhere...  even a first approximation (neglecting the moon
and so on) was very close.

- Jim

    re .1
    I believe that the stick should not be straight vertical.  It must
    be tilted according to your latitude.  I'm sorry I don't know what
    the relationship is.
    
    Carl

    Offhand it seems as though the arm of the sundial should point to Earth's
    celestial North or South, i.e. vertical at the poles and horizontal
    at the equator. Where's my Encyclopedia Britainica when I need it?

    At the Boston Museum of Science they have some exhibits in the Solar
    Energy Room that may help you.
    
    One is simply a globe of the earth with Boston on the top and oriented
    so that a line from Boston to the North Pole points North. This
    shows where the Earths surface is in the path of sunlight. 
    
    Another exhibit is a cube with different sundials on a number
    of faces.

    The library there usually has information on the exhibits. I am
    sure that you could find sundial information in any library, but
    the museum may have some information on very unique types.

    I also recall that "Scientific American" had a good article a few
    years ago. The obvious problems are the varying length of the day
    (which doesn't expand or contract symetrically on noon) and daylight
    savings time. Perhaps a clock driven sundial :-). 
    
    /joel
    

    A good source of information, (that I used to build a simple sundial),
    is   Sundials-Their Theory and Construction
         Albert E. Waugh
         a Dover book, what else :-)
    carl
    

    From the Golden Home and High School Encyclopedia:
    
 "The gnomon is the shadow-producing part of the sundial and is usually
    somewhat triangular in shape, with its acute base angle at the center
    of the dial and its hypotenuse parallel to the earth's axis. This
    means that the gnomon must point to the north, and the acute base
    angle must equal in terms of degrees the latitude of the exact spot
    on the earth to which the sundial is fixed."
    
    

Although I can't remember the source of this memory, I recall
see a picture of very pretty sundial which was not like your
average 'stick in the ground' style of time keeping device.
Instead the surface which the shadow fell on was a curved band
which would have encircled the shadow creating rod if it had gone
360 degrees around.  As it was, it probably went less than 180.

I think that this whole apparatus was point up at the angle
described in .-1.  There might have been some additional bands
which allowed you to know what day of the year it was.  Although
I might be wrong, I think the normal flat sundial is only
right at noon (and during 2 days of the year) this one I
believe compensated for this.

I would guess that I saw it described in the Amatuer Scientist
column in scientific American several years ago.

I think over 10 years ago, there was an article in Astronomy magazine on
a sundial which was a small mirror at the right angle set in the author's
window.  His ceiling had a curvilinear coordinate system laid out with ribbons
from which you could read the correct time knowing the date.

As for simple sundials, the Dover book '100 Great Problems of Elmementry
Mathematics', #83 is the 'Problem of the Sundial'.

If you consider the gnomon the hypotenuse of a right triangle, with base
running north/south and height straight up, the angle between the base
and hypotnuse will be your latitude.  The angle the gnomon's shadow
will make with the base (in a horizontal plane) will satisfy

	tan(shadow_angle) = sin(latitude)*tan(time),

where time is the rotation of the earth in radians since local noon.

The book has lots of neat problems, navigational, astronomical, etc, some
of which are not that trivial; well worth looking thru.  It also contains
a solution of the shodow curve, which you could use to devise a sundial
via mirrors as mentioned above.

- Jim

The mirror sundial sounds neat!  I just saw a sundial of the
type which I was trying to describe in a previous reply.  It
is in front of a Boston University building down on Comm. ave
right before Kenmore sq.  (the building in front of the nickalodeon
theatre.)