sundial mathematics

A study of sundial calibration for eight selected altitude dials

Mathematical diagrams for the circular altitude dial.

This dial has circular graduations for dates. The gnomon is fixed, and the hour point on a circle is where the nodus shadow falls. A series of calculations for coordinates (X’, Y’) give the calibration for the whole dial. These calculations are by numerical iteration on computer.

Altitude dials are an intriguing type of sundial. There are many individual types, and they mainly date from around the 1600s or earlier. Our own design for a full-scale altitude sundial for the garden is a circular dial. It follows the same working principle as a circular brass dial in The Augustine Museum in Germany. We decided to investigate the calibration of eight different altitude dials, including the one from the museum, and we have now completed a study of their mathematics. We submit the report of our study in full for wider interest here: A Study of Altitude Sundial Calibration.

 

 

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A circular altitude sundial at full-scale for the garden

Altitude sundial in the garden

The altitude sundial in sunshine on 21st July 2026. The dial reads 3:50 pm, which is only 4 minutes difference from clock time, with corrections made for BST, Longitude, and EOT. The reading point is the tip of the shadow, which is the shadow of the gnomon nodus. 

The altitude sundial is an intriguing instrument. It relies on measuring the height of the sun to tell the time. It also requires the user to adjust for the time of year before making a reading. There are many altitude sundials and there is one particular one with a circular dial, from around the year 1600, which we find very interesting. We have now designed and built one like it at full-scale for the garden. Continue reading

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Inverkeithing Mercat Cross – sundial restoration complete

Inverkeithing Mercat Cross, the capital stone, cube sundial, and unicorn fully restored.

The Inverkeithing Mercat Cross is a tall stone column in the centre of the town. The capital stone, cube sundial, and unicorn sculpture are at the top of the column. There has been no ‘Cross’ as a Christian symbol on the column since the Reformation. Photo: Graciela Ainsworth.

The Inverkeithing Mercat Cross is a tall and handsome piece of historic town architecture. The Mercat Cross in Scottish towns was a symbol of licensed authority to trade in goods, and the trading place was at the Cross. The sundial would be a 17th century addition, now fully restored, together with the heraldic capital stone and the symbolic unicorn of Scotland.

The sundial is a cube stone which is ideal for a prominent place in a town. It is visible from all sides and it shows the time at all hours of the day. Continue reading

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Haddo House cube stone sundial restored and reading solar time

Haddo House stone cube sundial

This sundial cube stone has fine carving detail. Each of five faces can read the time. The horizontal dial carved in the stone on top indicates an ornamental piece for a garden. A cube set on a tall pillar or on a wall will have the side dials only. Photo: Graciela Ainsworth.

This cube stone sundial stands in the garden of Haddo House, the National Trust for Scotland property in Aberdeenshire. When we examined it at the workshop, it had broken and missing gnomons and moss obscured the dials. After restoration the dials can now read solar time properly again.

A cube dial has the mathematical attraction of reading time by the sun on south, east, west, and north faces. As a result, as the sun moves round, the cube sundial is an excellent device for showing time Continue reading

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A modern altitude sundial after Erasmus Habermel

A modern circular altitude sundial after Erasmus Habermel of Prague

Modern reconstruction of a 16th century altitude sundial for studying its mathematics and accuracy. An altitude dial uses the height of the sun to find the time. The dial is double-sided with the two half-years back-to-back.

Erasmus Habermel was a renowned scientific instrument maker working in the court of Emperor Rudolf II in Prague in the 16th century. He made instruments for all disciplines including astronomy and navigation. One of his sundials is an altitude dial in the clocks collection of the Augustine Museum in Freiburg im Bresgau*. It is an elegant circular dial and the mathematical construction is intriguing. Continue reading

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Calibrating a new obelisk sundial in stone

Inclined test dial facing east

This dial stone is a test piece for the new obelisk sundial, and is on a board at the correct angle for one of the faces of the obelisk. It faces East and inclines 61° to horizontal so the shadow of the temporary plastic gnomon will show the time.

Obelisk sundials carved in stone were popular in Scotland in the 17th and 18th centuries. Quite recently stonemasons in Fife have done the work to carve a new obelisk sundial for one of their clients. Its dials have remained blank for mathematical calibration. Macmillan Hunter are responsible for the calibration and have supplied detailed dial instructions and drawings for the mason to use together with the required set of geometric bronze gnomons. Continue reading

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Historic Scottish Stone Sundial Restored

Scottish stone sundial restored

The octagonal base and baluster shaft support a carved cube stone. Before restoration the stones were covered in moss lying on the ground in the owner’s garden. The bronze horizontal dial is new.

Stone sundials were common in Scotland in the 17th and 18th centuries. They stood in the gardens of great houses, and in public places. The geometric form of the sundial would reflect the popular interests of the day in science and mathematics, and many displayed complex multiple dials. Originally this particular sundial may have belonged to a country estate near Falkirk. The present owner has had it for many years and in 2023 decided to restore it. Continue reading

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An armillary sundial in early morning

An armillary sundial in early morning

This is our new sundial design. It stands in our garden among the plants. It is early morning and the sun has just reached the centre band where you read the time.

An armillary sundial has a wonderful form. It has mathematical rings creating beautiful shapes with circles and lines. The sundial in the photo has just begun to catch the early morning sun in the garden.

The sundial gnomon is the slanting rod in the centre of the rings, and the wide sweeping band is where you can read the time. These features are the same as other armillaries, but our design has special secrets. Continue reading

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Modern techniques for a replica sundial exhibit

A first trial assembly of the gnomon and base plate in position on the dial

Details for making the sundial came from archive records and photos. This image shows the design and successful first assembly of the new parts.

The replica of a 19th century lighthouse sundial we are making will be part of a mobile exhibition. It is progressing well. Lighthouse sundials were made by skilled instrument makers, who worked mainly with hand tools. We are using modern techniques for cutting out the metal and etching the intricate design on the dial plate. The picture shows our first trial to assemble the dial and gnomon parts. Continue reading

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What is the declination of the wall for a sundial

The diagram illustrates a method for calculating wall declination.

Knowing the declination of the wall (WD) is important for designing a sundial. The basic calculation is azimuth angle (AA) minus protractor angle (PA) plus 90°. You can repeat your measurements for accuracy. Do ask for help if you are unsure.

“What is the declination of the wall for a sundial?” A stone sculptor asked us just this question recently . It is a good question because you must know the declination if you are making a sundial that is accurate for the wall. The sculptor was not confident about their own measurements and calculations so they asked us for help. Continue reading

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