The Huggett Heliochronometer
A description of the design and construction of a garden sundial that reads clock time to within a minute or two
an article decribing this project appeared in the journal of the British Sundial Society, the Bulletin, in September 2017

The link to the page that describes my second heliochronometer is at:
http://bit.ly/The_Huggett_Heliochronometer_Mark_II

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C – Are there different types of sundials?

Yes, sundials are classified into a number of different types. These are mainly defined by the plane of the dial plate – the part on which the time scale is mounted.

There are any numbers of varieties of each type depending on the features incorporated into their designs.

D – What type of sundial is the Huggett Heliochronometer?

The Huggett Heliochronometer is an equatorial sundial.

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F – Why did you choose an equatorial sundial?

Because of the exact analogy between an equatorial sundial and the real Earth, the graduations on the time scale have equal spacings – one hour is always 15 degrees of the circle on which the time graduations are marked.

This is because 360 degrees of movement by the Sun equals 24 hours, and so any hour on the time scale of an equatorial sundial always spans 360/24 = 15 degrees.

This means that corrections for longitude, the Equation of Time and Daylight Saving Time, as described below, can be made by simply rotating the dial plate/time scale.

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G – How can a garden sundial be made to read clock time?

Most garden sundials are incorrect in relation to clock time. They are essentially garden ornaments.

In order for a sundial to correspond with a properly adjusted clock, it must be corrected to take account of five factors.

These factors are each described in detail below, but a list of them is as follows:

1 – The latitude of the sundial.
2 – The
longitude of the sundial.
3 – The
north-south orientation of the sundial.
4 –
The Equation of Time The EoT.
5 –
Daylight Saving Time.

The Huggett Heliochronometer incorporates corrections for all of these.

After explaining the above factors and how my sundial adjusts for them, the notes below consider the design implications for the sundial as a result of the
Declination of the Sun.

From this emerges an
Interesting quiz question for any quiz setters.

I then consider the
Accuracy of the sundial in practice.

Finally, I explain some of the
Practical issues involved in constructing the Huggett Heliochronometer.

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A – How is time measured?

A second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

That definition allows an equally precise description of a minute, and then of an hour.

But what of a day?

For practical purposes, the duration of a day must continue to reflect the period of light and the period of darkness that was familiar to our earliest ancestors.

This 'solar day', at any point on the Earth, is the interval between the Sun being at its highest point in the sky (its zenith) and the next occasion when that occurs.

Adjustments are required to our super-accurate clocks to ensure that our days, months and years remain synchronised with the rotation of the Earth and its movement around the Sun.

The beauty of a sundial is that, unlike our mechanical and digital clocks, it directly reminds us of that connection – the fundamental link between our lives, our planet and the Universe.

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B – What is a sundial, and what is a heliochronometer?

A sundial is an instrument that uses sunlight to indicate the time.

The term 'heliochronometer' simply refers to a sundial that can show the same time as would appear on a properly adjusted clock – within the margins of error of the sundial.

This distinction between a sundial and a heliochronometer implies that the time measured by a sundial is not necessarily the same thing as the time measured by a properly adjusted clock.

That is indeed the case. Solar days, as known by our ancestors, are not of a constant duration.

Sundials that incorporate no corrections read solar time, and solar time only corresponds to local clock time, at any given location, on four occasions each year.

Currently those coincidences occur on 15th April, 15th June, 1st September and 24th December.

At other times of the year, solar time and local clock time can differ by up to sixteen and a half minutes.

The reason for this is that clock time assumes every day to be exactly 24 hours. The Earth’s orbit around the Sun is not perfectly circular, however, and the axis of the Earth is tilted in relation to the plane of its orbit. These and other factors mean that the duration of a day, as measured by the time interval between one noon and the next, differs throughout the year, everywhere on Earth.

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E – What is an equatorial sundial?

An equatorial sundial has its dial plate aligned to be parallel to the plane of the Earth’s equator.

Imagine that you hold up a model of the Earth on a sunny day such that it is exactly aligns with the orientation of the real Earth. This is to say that a straight line joining the north and south poles of the real Earth is parallel with the same line on your model, and the plane of the equator on your model is parallel with the plane of the equator on the real Earth.

The sunlight will fall upon your model in exactly the same way as it falls upon the Earth.

Imagine now that you replace your model of the Earth with just a disc representing its equatorial plane and a rod representing the straight line between the north and south poles.

That disc is the same thing as the dial plate of an equatorial sundial.

The rod in your model is called a gnomon and will cast a shadow onto the dial plate/time scale that matches the location of the Sun in the sky. It hence indicates the time of day.

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1 – Latitude

As mentioned above, the dial plate of an equatorial sundial must be aligned to be parallel with the plane of the Earth's equator.

The latitude of my garden is 50.94 degrees north.

In order to align the dial plate of my sundial to be parallel to the plane of the Earth's equator it must be set at (90 – 50.94) degrees to the horizontal. For practical purposes that is 39 degrees.

The geometry may be more easily understood from the diagram on the right, rather than by explanation.

If the diagram is not clear to you, imagine a cross section of the Earth, and imagine yourself standing on the surface of the Earth, looking horizontally in front of you.

Imagine a line from the centre of the Earth to the equator. Now imagine a line from the centre of the Earth to you. The angle between those two lines, measured at the centre of the Earth, is your latitude.

If you are standing on the equator, that angle is zero, and your latitude is zero.
If you are standing at the true North Pole, your latitude is 90 degrees.

As you are looking horizontally in front of you, your line of sight is at right angles to the line joining you with the centre of the Earth.

To look along a line that is parallel with the plane of the Earth's equator, you would need to look up from the horizontal by 90 degrees minus your angle of latitude.

The angle of 90 degrees minus your latitude is sometime called your co-latitude.

My sundial is designed to be permanently positioned in my garden and so some elements of the design do not need to be adjustable.

The co-latitude of 39 degrees is therefore built into the structure.

Clearly, it would be possible to incorporate a mechanism that would permit the dial plate to rotate in the vertical plane and thus allow the sundial to be used at other latitudes.

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2 – Longitude

 


The view towards the Thames from the Royal Observatory at Greenwich



The Prime Meridian at Greenwich, England

Tourists at the Greenwich Royal Observatory queuing to be photographed on the Prime Meridian.
Presumably few have grasped the comic irony of queuing to stand on a non-existent, abstract, mathematical concept – the ultimate British queue!!

Clock time in relation to where I live is either the mean solar time at Greenwich (GMT) or that time plus one hour (British Summer Time or BST).

Whether my clock time is BST or GMT, however, it still relates to the Greenwich (Zero or Prime) Meridian.

If one imagines the Sun apparently rising in the east and setting in the west, then a given solar time, say solar noon, occurs later as one moves west.

A sundial in the UK must compensate for its longitude, therefore, if it is to reflect the solar time at the Greenwich Meridian.

My garden is at a longitude of 1.178 degrees west.

As explained above, 15 degrees on the circle of the dial plate represents one hour. This is because the Sun appears to move 15 degrees across the sky in one hour.

1 degree of longitude, therefore, represents 60/15 = 4 minutes of solar time.
1.178 degrees represents 4.7 minutes.

My garden lies west of Greenwich, so, when it is solar noon at the Greenwich Meridian, it is 4.7 minutes before solar noon in my garden – the Sun has yet to reach solar noon in my garden on its apparent journey from east to west.

To put it another way, when the solar time is 12:00 noon at the Greenwich Meridian, the solar time in my garden is just after 11:55 am.

I want my sundial to read 12:00 noon when it is 12:00 noon at the Greenwich Meridian and so the time scale must be rotated to make my sundial read 4.7 minutes later than local solar time.

This is very simply effected with an equatorial sundial. To correct for the longitude of my garden, the time scale is positioned by a permanent anti-clockwise rotation (as viewed from the top) to read, as near as can be estimated, 4.7 minutes later than local solar time.

Due to the graduations on the time scale having equal spacings, it would be straightforward to adjust for other longitudes by unattaching the time scale from the dial plate annulus and reattaching it with an appropriate rotation.

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3 – North-South Orientation

In order for the orientation of my sundial to match that of the Earth, as described above, the gnomon (the rod that casts the shadow on the time scale) must be aligned to point towards geographic (true) north.

If you looked along its alignment on a starry night, it would aim at the star called Polaris, the North Star.

There is a difference between geographic north and magnetic north that changes with time and location on the Earth.

Nm in the diagram on the left is magnetic north and Ng is geographic north.

The angle between them is called the magnetic variation or magnetic declination.

If magnetic north is east of geographic north then the magnetic declination is consisted as positive.
If magnetic north is west of geographic north then the magnetic declination is consisted as negative.

The current (2017) magnetic declination of my garden is 0.9 degrees west.
Using the above convention, this is minus 0.9 degrees.

The sundial's alignment is achieved by having a compass rose on its baseplate with its north-south axis parallel to the longitudinal axis of the sundial.

A red line is marked on that rose with a black line running along its centre.

As can be seen in the photograph on the right, that red line is rotated by 0.9 degrees anticlockwise relative to the longitudinal axis of the sundial.

A large (100mm), free floating, compass needle can be placed on a pivot at the centre of the rose.

By rotating the whole sundial, the compass needle can be aligned with that red line, causing the longitudinal axis of the sundial to point towards geographic north.

This method worked adequately in practice, although could only be used when the air was absolutely still, as a breath of wind moved the needle.

It was possibly more luck than judgement that an accurate north-south orientation was achieved using a compass needle. I understand that this is not the preferred method for diallists.

Due to the accuracy of modern clocks, it is, of course, possible to align a dial to north by making all other relevant adjustments to the dial and then orientating it such that it reads the expected time.

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4 – The Equation of Time – The EoT

To recap on what I have written above, sundials that incorporate no corrections read solar time, and solar time only corresponds to local clock time at any given location on four occasions each year.

Currently those coincidences occur on 15th April, 15th June, 1st September and 24th December.

At other times of the year, solar time and local clock time can differ by up to sixteen and a half minutes.

The reason for this is that clock time assumes that every day is exactly 24 hours. The Earth’s orbit around the Sun is not perfectly circular, however, and the axis of the earth is tilted in relation to the plane of its orbit. These and other factors mean that the duration of a day, as measured by the time between one noon and the next, differs throughout the year.

The difference between solar time and clock time on any day of the year is described by what is referred to as the ‘Equation of Time’ or EoT.

Probably the easiest way to understand this is by looking at a graph of the EoT on the left.

The graph shows the number of minutes that must be added to, or subtracted from, the reading of a sundial to correct for the varying lengths of the solar day throughout the year.

The difference between clock time and solar time on any given day is not the same each year. It is close enough from year to year, however, to be of little concern when reading recently constructed sundials. It will, nevertheless, have an impact over decades and centuries.

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How does the Huggett Heliochronometer take account of the Equation of Time?

A graph of the equation of time is mounted above the dial plate annulus – as shown in these photographs.

The dial plate annulus, and hence the time scale attached to it, can rotate in the plane of the dial plate, relative to the rest of the sundial's structure.

An acrylic pointer is attached to the dial plate annulus such that it rotates with the the dial plate annulus.

By rotating the dial plate annulus to align the slot in the acrylic pointer with the current date on relevant time curve, the dial plate annulus, and hence the time scale, is moved to compensate for the Equation of Time.

In the photograph below, the adjustment scale is set for the 15th of March – although would also be correct for the 15th of January.

Creation of the EoT and Daylight Saving Time Adjustment Scale required:
* a basic knowledge of trigonometry and polar co-ordinates;
* a list, found on the Internet, of the difference between clock time and solar time for each day of a recent year;
* a spreadsheet to do the calculations and
* a graph program to plot the results.

An honourable mention for that graph program is well deserved. It is excellent; it is free and can be found at https://www.padowan.dk/.

The principle on which the graph plot is based is straightforward:

As explained above, the Sun appears to move 15 degrees across the sky in one hour. Each hour on the sundial's time scale, therefore, occupies a 15 degree arc of its circle.

60 minutes of time equals 15 degrees on the sundial's time scale, and so 1 minute of time equals 15/60 = 0.25 degrees.

If, on a given day, solar time differs from local clock time by 10 minutes, then to correct the sundial reading to local clock time, the dial plate/time scale must be rotated by 10 X 0.25 = 2.5 degrees.

The relevant point for the day on the appropriate EoT Adjustment Graph is calculated such that aligning the slot in the acrylic pointer with that point will cause the dial plate/time scale to rotate by the required 2.5 degrees. See the section on Daylight Saving Time, below, for the reason why there are two such graphs.

The lines that form the time curves are just less than 3 mm wide, as is the slot in the acrylic pointer.

That seems quite large for high accuracy, but I found that estimating the centre of the graph line and crossing that with the centre of the slot in the pointer was easier to see, and just as accurate, as using very narrow graph lines and a very narrow slot in the pointer.

I felt sure that someone must have previously thought of this way of directly correcting for the EoT. I had never seen it in real life or on the Internet, however, and so, for me, this was an original idea – conceived on 20th November 2016, should the date be required for historical research!

A second 'original' idea conceived on the same day was about how to correct for Daylight Saving Time.

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5 – Daylight Saving Time

In the UK, Daylight Saving Time is called British Summer Time (BST) and occurs from 2.00 am on the last Sunday in March until 2.00 am on the last Sunday in October.

During the period of BST, clock time is one hour later than Greenwich Mean Time (GMT).

The Huggett Heliochronometer adjusts between GMT and BST by simply having two EoT Adjustment Graphs set 15 degrees, or one hour, apart – as shown in the photographs above.

The two graphs are not currently marked to highlight the dates for the changes between GMT and BST (the last Sunday in March and in October), but they could be.

As described below, I intend to replace the EoT and Daylight Saving Time Adjustment Scale with one printed on vinyl – for improved durability. I will use that opportunity to make some changes to the graphics. Those changes will include marking the dates for the GMT/BST changes and rendering in a different colour or style the sections of each EoT Adjustment Graph that are never used.

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The Declination of the Sun

Most people in the UK are aware that the Sun appears high in the sky in summer and low in the sky in winter.

The angle that the Sun appears to lie above or below the plane of the equator is called its declination.

This figure is considered as positive when the Sun appears north of the plane of the equator (above in the northern hemisphere) and negative when it appears south of the plane of the equator (below in the northern hemisphere).

The graph below shows the variation in the Sun’s declination during the year.

This variation in declination happens because of the tilt of the Earth’s axis as the Earth orbits the Sun.

The northern hemisphere is inclined towards the Sun between the vernal (spring) equinox and the autumnal equinox, and the southern hemisphere is inclined towards the Sun between the autumnal equinox and the vernal equinox – hence causing our seasons.

The Sun appears to cross the equator at the equinoxes – the days when day and night are of equal length.

The angle of declination is zero at the autumnal and vernal equinoxes when the Sun appears to lie on the celestial equator.

As described above, the dial plate of the Huggett Heliochronometer, because it is an equatorial sundial, is parallel to the plane of the Earth's equator.

From the autumnal equinox (22nd September in 2016) until the vernal (spring) equinox (20th March in 2017), therefore, sunlight falls upon the underside of the sundial's dial plate. For the rest of the year, sunlight falls upon the upper side of the sundial's dial plate.

The Huggett Heliochronometer has a time scale positioned at right angles to the plane off the dial plate, attached to the inside surface of the dial plate annulus. This means that sunlight can shine upon that time scale from either above or below the dial plate.

The same scale can thus be easily read from the top of the sundial on sunny days, all the year round.

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Reading the time scale on days close to an equinox

The photograph above illustrates how the time scale protrudes above and below the dial plate. The upper or lower sections can be read on days close to an equinox when the Sun is near the plane of the dial plate.

The photograph was taken just before the spring equinox and so also illustrates the shadow cast by the dial plate at that time.

The section above concerning the declination of the Sun explains why the Sun will be in the plane of the dial plate on days close to an equinox.

The Huggett Heliochronometer is an equatorial sundial with a time scale positioned at right angles to the plane off the dial plate. The time scale is attached to the inside surface of the dial plate annulus.

The dial plate annulus, however, forms a complete circle. This means that the time scale will be in the shadow of the dial plate on days close to an equinox.

Some equatorial sundials of a similar design have part of their dial plates missing to avoid the Sun being obscured by the dial plate on days close to an equinox.

That does not usually affect time readings because the missing part of the time scale would represent times either side of midnight when the Sun would not be shining at most latitudes.

The design of the Huggett Heliochronometer, however, requires the full equatorial circle to be in place as the mechanism for two of the five corrections listed above, The Equation of Time and Daylight Saving Time, is mounted on that section of the dial plate.

As can be seen in the photograph, the problem is partly resolved in the Huggett Heliochronometer by having a time scale that is deeper than the dial plate.

This time scale cannot be a full circle, however, because the time scale would be obscured by its own shadow on days close to an equinox.

In order to avoid that problem, a section of the time scale has been omitted.

The time scale runs from 6.00 am to 9.00 pm, and so most of the absent section of the time scale, for much of the year, would represent the hours of darkness.

The shadow cast by the time scale onto itself means that on days close to an equinox, the time can only be read between 9.00 am and 6.00 pm.

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An interesting quiz question

If you are ever setting a quiz, you could employ the photograph above – the one reproduced on the right.

The question could be: 'This is a photograph of the time scale of an equatorial sundial. At approximately which two times of year could the photograph have been taken?'

The full answer is: 'At around the time of the vernal (spring) equinox – between mid-March and early April in the northern hemisphere, and at around the time of the autumnal equinox – between late September and mid-October in the northern hemisphere.'

The shadow of the dial plate falls fairly centrally on the time scale with a band of sunlight at its top and bottom. This means that the Sun is reasonably well aligned with the plane of the dial plate. That would only occur on days close to an equinox.

The photograph was actually taken on 15th March 2017.

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Accuracy

There are many factors that bear on the accuracy of a sundial. These are related to accuracy in setting the parameters discussed in these notes.

I will not list the theoretical effect that an error in setting each parameter might have. Explanations of these are available on the Internet.

I have simply preriodically compared the time displayed by the sundial to the time shown by my watch.

The accuracy has been around one minute, and certainly no more then two minutes.

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Practical issues involved in constructing the Huggett Heliochronometer

How is the time scale positioned in relation to the dial plate?

If the sundial was simply reading local solar time with no corrections for EoT, longitude or Daylight Saving Time, then 12:00 noon on the time scale would align with the longitudinal axis of the dial plate at its northern (lower) end.

This is where the gnomon would cast its shadow at solar noon, when the Sun is at its highest point in the sky, because all shadows align to the north at solar noon.

As described above, however, the longitude correction requires the scale to be permanently rotated anti-clockwise by 4.7 minutes to compensate for local solar time being earlier than solar time at the Greenwich Meridian.

If longitude was the only correction, then the time scale at the northern (lower) end of the longitudinal axis of symmetry of the dial plate should be positioned to read 12:04.7 pm.

The EoT and Daylight Saving Time adjustment scale is on the longitudinal axis of the sundial, however, with the GMT graph set half an hour (7.5 degrees) clockwise of the axis of symmetry and the BST graph set half an hour (7.5 degrees) anti-clockwise of the axis of symmetry.

If the dial plate was positioned such that the GMT and BST graphs lay symmetrically on either side of the longitudinal axis of the sundial, and there were no other corrections, the time scale at the northern (lower) end of the longitudinal axis of symmetry of the dial plate should be positioned to read 12:30 pm.

If one then adds the longitude correction to this, the time scale at the northern (lower) end of the longitudinal axis of symmetry of the dial plate should finally be positioned to read 12:34.7 pm.

It is easy to mark the longitudinal axis of symmetry of the dial plate annulus and the corresponding lateral axis at right angles to it.

In attaching the time scale to the dial plate annulus to take account of both the above adjustments longitude and the positioning of the EoT/Daylight Saving Time adjustment scale the cardinal points on the dial plate annulus must align with times on the time scale as follows:

North: 12:34.7
East: 18:34.7
West: 06:34.7

These locations allow the time scale to be aligned with the dial plate annulus at three points along its circumference in order to confirm its accuracy and positioning.

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How is the sundial levelled to the horizontal?

The sundial stands on three feet that are adjustable using screw threads. One of these adjustable feet is shown in the photograph on the right.

Having three points of support allows the sundial to stand on uneven ground. It also simplifies the levelling process.

The level is checked with a spirit level designed for record player turntables – as illustrated below.

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What is the mechanism that allows the dial plate to rotate?

The dial plate rides on stainless steel ball bearings that are located in a groove between the moving section of the dial plate and a static support section of the dial plate.

The picture below shows the groove containing ball bearings, prior to final assembly.

The dial plate is supported in its position, and guided in its rotation, by nylon wheels on stainless steel bearings – as shown below. These were designed to support the sliding doors of a shower.

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How did you draw the scales and the compass rose?

They were all designed and drawn on the computer (the main compass graphic was taken from the Internet) and then printed on photo quality paper.

A section of the time scale is shown below. The time period between graduations is two minutes.

A small, red alignment mark can be seen at the top of the scale at 12:34.7 pm. This aligns with the longitudinal axis of the dial plate, as explained above.

Fortunately, I discovered that printer ink is not soluble in varnish. Also that ‘Ronseal Crystal Clear Outdoor Varnish’ does what it says on the tin in that it is perfectly transparent.

Before embarking on the main project, I built a small scale prototype sundial of the same design to help me understand the physics, the design and construction issues and the materials.

Unfortunately, both sundials have demonstrated that scales printed on photographic paper are insufficiently durable. Although printer ink does not dissolve in varnish, and so they seem well protected from rain, it fades in sunlight.

I intend, therefore, to replace all the printed scales with ones that are commercially printed onto vinyl.

I am happy with the design of the time scale and compass rose, but will use the opportunity provided by scale replacements to enhance the graphics of the Equation of Time and Daylight Saving Time Adjustment Scale.

Those enhancements will include:
*Highlighting the dates for the changes between GMT and BST (the last Sunday in March and in October).
*Rendering in a different colour or style the sections of each Equation of Time Adjustment Graph that are never used.
*Increasing the number of date arcs to improve accuracy in setting the acrylic pointer.

How did you avoid the shadow cast by the gnomon being too wide, and how did you keep a long, thin gnomon straight?

Looking from the Earth, the Sun has an angular diameter of 0.5 degrees – an angle equivalent to 2 minutes on the time scale of an equatorial sundial..

For any equatorial sundial that uses the direct shadow of a gnomon, therefore, this shadow will have an umbra and a penumbra because the rays of sunlight that cause the shadow come from every point on the 0.5 degrees of the Sun's apparent width.

It is possible to estimate the location of the centre of the shadow and use that for a time reading. It is useful to begin, however, by making the shadow of the gnomon as narrow as possible.

Reducing the thickness of the gnomon is one obvious approach to making its shadow as narrow as possible, but it is difficult to keep a very thin rod straight.

My sundial uses 0.8 mm stainless steel wire for the gnomon that is tensioned between its support points. This means that the gnomon can both be thin and totally straight.

One minute on the time scale is 0.85 mm in width.

The width of the shadow cast by the gnomon onto the time scale, therefore, is around the equivalent of two minutes, with the umbra of the shadow spanning the equivalent of about a minute.

The shadow of the gnomon at around 14:09 is shown in the photograph below.

A guitar machine head made an excellent tensioning mechanism for the gnomon wire as shown in the photograph below.

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How did you bend the wood for the time scale?

The pine stripwood from which the time scale is made was immersed in water overnight.

It was then bent around the former, as shown in the photograph on the left, by pressing in onto the former, a section at a time, using a hot domestic iron.

Each section was screwed to the former with the wooden blocks shown.

The circular sections of the former were made from the wood that had been cut from the centre of the dial plate. They were therefore of the correct diameter for the time scale.



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How did you cut all the circles and arcs of various sizes needed for the sundial?

I made this very simple jig that allowed my jigsaw to cut circles.

The screw, shown in the photograph, attaches the jig to the centre of any circle to be cut and acts as a pivot for the jigsaw to rotate around. Any diameter of circle can be cut by drilling a hole for the screw at an appropriate place on the bar.

Mine is not a scrolling jigsaw, but that does not seem to matter if the blades used have a small depth, front to back.

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Copyright – Brian Huggett – 2017

heliochronometer@huggett.info

Brian Huggett is a retired mental health social worker. He enjoys growing vegetables; performing as a singer/guitarist and writing under the pen name of Swan Morrison:
https://www.linkedin.com/in/shorthumoursite/.
He is also a member of the British Sundial Society:
http://sundialsoc.org.uk/.

The link to the page that describes my second heliochronometer is at:
http://bit.ly/The_Huggett_Heliochronometer_Mark_II

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Copyright Notes:

All the images used on this page are my own copyright with the exception of the four cited below:
My images may be used for non-commercial purposes with the attribution:
Image by Brian Huggett (http://www.short-humour.org.uk/Heliochronometer/Heliochronometer.htm).

Details of copyright in respect of the other mages used on this page are as follows:
1 - The original compass rose graphic prior to amendment:
This was a free image obtained from:
http://clipartix.com/compass-clip-art-image-18691/
2 - The graphic illustrating magnetic declination:
Attribution: By odder [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/), CC BY-SA 2.5 (http://creativecommons.org/licenses/by-sa/2.5) or GPL (http://www.gnu.org/licenses/gpl.html)], via Wikimedia Commons.
Details of licence at:
https://commons.wikimedia.org/wiki/File:Magnetic_declination.svg.
3 - The graph of solar declination:
Taken from the following webpage:
http://www.itacanet.org/the-sun-as-a-source-of-energy/part-3-calculating-solar-angles/.
The site states that the image is copy-left and freely reproducible.

4 - The Equation of Time graph:
Attribution: The British Sundial Society.
Taken from the following webpage:
http://sundialsoc.org.uk/old/HDSW.php.

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