This week, build an astrolabe’s tympan—latitude—plate to represent the arcs for Unequal Hours
Tympan Plate: Unequal Hours
Throughout the year, the number of daylight hours in a day changes based on the time of year and an observer’s latitude. The Earth sits at an axial tilt of about 23.45° relative to its plane of orbit. The Earth’s axial tilt—also known as the obliquity—causes the seasons because throughout the year different hemispheres are tilted either towards or away from the Sun. When the North Pole is tilted towards the Sun, it is summer in the Northern Hemisphere (and winter in the Southern Hemisphere), and when the South Pole is tilted towards the Sun, it is winter in the Northern Hemisphere (and summer in the Southern Hemisphere).
At higher latitudes, the winter months have a shorter “day” with fewer daylight hours while in the summer months the number of daylight hours is greater. This culminates at the solstices. The winter solstice represents the shortest day, with the fewest number of daylight hours, while the summer solstice is the longest.
Unequal hours are a type of time keeping. As opposed to equal hours where the day is split into 24 equal hours—each 60 minutes long—unequal hours are based on the number of daylight hours, from sunrise to sunset. Hour one is when the sun rises, hour 12 is when the sun sets, and noon lies at the end of the sixth hour.
Historically, unequal hours have also been known as Roman hours as in ancient Rome. Unequal hours were one of the methods Rome used to measure the length of a day. When a Roman day began with the sunrise and ended with the sunset it was known as “dies naturalis” or a “natural day”. The natural day arose out of the introduction of the sundial to Rome, possibly as early as 263 BCE. In Pliny the Elder’s Natural History, he recounts the first public sundial arriving in Rome during the wake of the first Punic War. It was a spoil of war, captured from the Samnites and installed for public display atop the temple of Quirinus on the highest and most northern of Rome’s seven hills. However, the looted sundial had been made for the city of Catania in Sicily, south of Rome, and at an entirely different latitude, so the sundial was embarrassingly inaccurate. It was eventually replaced with a new sundial crafted to Rome’s particular latitude, but only decades later.
An earlier sundial had been previously developed by the Egyptians, with simple shadow clocks which divided the day into 10 hours and an additional hour at dusk and twilight. The simple sundial was made out of a T-shaped bar and would cast a shadow onto one of the twelve segments on the sundial based on the Sun’s position in the sky above.
Twelve was likely chosen because the traditional duodecimal (base 12) counting system mimics the total number of lunar cycles in a year and can be easily counted with the thumb along each of the three sets of finger joints in the hand.
A twelfth of the day—or hour—in a Roman natural day can vary based on the length of daylight hours due to the time of year and seasons. In the winter months, the sun only appears for a few hours, while in the summer it appears early in the morning and persists well into the evening. The inconsistent length of an hour is where the term “unequal hours” is derived.
Unequal hours began to fall out of favor in daily use by the late 14th century with the invention of mechanical clocks. Mechanical clocks work reliably all day, and, more importantly, all night, so the full day could be split into hours of equal length. However, unequal hours remained a common staple of astrolabes for religious purposes that relied on the position of the sun.
Salat times (aṣ-Ṣalāh, sometimes spelled Salah) in Islam are five daily prayers which Muslims perform at a relative, rather than exact, time of day. The prayers are one of the five fundamental pillars of Islamic faith and are at dawn before sunrise (al-fajr or fajr), shortly after midday (al-ẓuhr or Dhuhr), late afternoon (al-ʿaṣr or asr), sunset (al-maghrib or maghrib), and finally at nighttime before bed (al-ʿishā or isha). The timing of each prayer time is strict and dictated by the position of the sun, which in turn varies due to latitude and time of the year.
Unequal hours are an optional, but common, component of astrolabes. They provide a vital method to calculate sunrise and sunset needed for determining the relative position of the sun and prayer times. So, astrolabes crafted in Arabic countries were more likely to continue to include unequal hours even after the invention of mechanical clocks, which in turn influenced astrolabes constructed in other countries.
Unequal hours are used to split the length of daylight into 12 equal parts. This is less noticeable at lower latitudes, where the length of daylight does not vary dramatically throughout the year—always about 12 hours long—so each 12th segment will represent 1 hour. However, at higher latitudes, a day can be as short as 7 hours long or as long as 17 hours. So, as a result, at a solstice, a single 12th hour’s length can be as small as about half an hour or as long as an hour and a half.
On an astrolabe, unequal hours are represented visually by a series of 13 arcs along the bottom of the plate, beneath the horizon, separating 12 segments.
To construct the unequal hour arcs on an astrolabe, first, the base of the plate is formed with an arc to represent the horizon, for example, for an observer standing at 40° latitude (remembering to measure the angle from the pole and down 40°).
Projection lines are drawn (in blue) as two lines extending from the south pole until they intersect both the vertical equatorial plane and the edges of the circle representing horizon (an almucantar of 0°) for the observer’s latitude.
Where the horizon arc intersects the Tropic of Capricorn, Tropic of Cancer, and the equator is vital to form the arcs for unequal hours.
This creates a base plate with the tropics, horizon, and equator as well as the horizon for the observer’s latitude.
To begin creating the arcs for the unequal hours, first find where the horizon intersects the equator and tropics. For a valid range, the horizon needs to intersect all three concentric circles in the base plate.
The intersections will form six points along the horizon arc where the regions intersect the Tropic of Capricorn, Equator, and the Tropic of Cancer.
Each unequal hour arc is made up of three points formed by angles along the Tropic of Capricorn, Tropic of Cancer, and the Equator.
So, with the outermost circle which represents the Tropic of Capricorn, break up the bottom intersection line for the outer base into 12 equal segments. To divide the full 180° into 12 equal parts, split and draw a line every 15°. 15° is 1/12th of a full 365° circle and will represent a single hour.
Then, in the same way as the Tropic of Capricorn, break up the center line every 15° where it intersects the middle circle which represents the equator.
Finally, break up the top intersection line to where it intersects the innermost circle that represents the Tropic of Cancer.
Each intersection line should now be split up every 15° to create a series of points along the circles that represent the tropics and the equator.
The first unequal hour arcs lie along the horizon, representing 0° (sunrise) and 180° (sunset). These arcs lie on the horizon, representing the lowest and highest possible arcs.
Each remaining arc is created by connecting the points that represent the same degree between each circle. For example, find the points that represent 105° for each of the three concentric circles:
By connecting the three points, extend until it creates a circle that intersects the points along each concentric circle that represent 105°. The circle will help aid in determining how dramatically the arc curves.
Apply the same for the set of points between 105° to 180°. 
Because it is not possible to create a circle with a perfectly straight line as part of its circumference, at 90°, draw a straight vertical line. In unequal hours, this represents the middle of the day. Unlike how the time of day changes when the sun rises or sets, noon is always at the middle of the day.
Finally, find the sets of points for the left side that represent 0° to 75°.
The unequal hours arcs are formed by only displaying the section of the circles that are contained within the base plate and are below the horizon arc. This is done to avoid overlapping with the azimuth and almucantar arcs that are contained within the horizon. If some arcs for twilight hours are included in the astrolabe, below the horizon, it is possible to have some overlap between where arcs for the unequal hours and the twilight hours are engraved.
So, finally, for an observer at 40°, about half way above the equator towards the North Pole, the unequal hours curve noticeably.
It is important to note that since the Earth’s obliquity is 23.45°, then there are regions on the planet that throughout the year can be entirely in darkness or entirely in daylight for days on end. These polar regions—above 66.55° and below -66.55° (where |23.45° – 90°| = 66.55°)—are regions where a valid range of unequal hours cannot exist on an astrolabe because daylight varies so dramatically. So, latitudes above 66.55° will not have a valid series of unequal hour arcs.
It is possible to see why this region is an invalid range with how unequal hours are created with spherical projection. Unequal hour arcs required six points where the horizon intersected with the Tropic of Capricorn, Tropic of Cancer, and the equator. However, visually, the latitude at 66.55° is the highest latitude where the horizon can still intersect all three concentric circles that form the base plate. Above this latitude, the horizon would lie entirely within the base plate, so it would no longer intersect all three concentric circles, since it does not touch the outermost circle. 
Therefore, the last possible position to contain valid unequal hours is 66.55° where the arcs that form unequal hours are the most dramatic.
However, at lower latitudes, the difference in daylight hours during the year will not be as noticeable, like at 5° latitude, where each line is almost a perfectly straight line: 
Only at the equator—0°—is the length of the day equal throughout the year.
The twelve segments for the unequal hours are ordered (1-12) starting from the east to the west in a clockwise order. 
The cardinal directions are based on the position on the intersection points of the horizon. North lies within the base plate, at the midpoint of the horizon arc. However, while south does not lie within the plate its general direction can be found along the meridian where its projection lies:
East and west are found in clockwise order from the positions of north and south: north, east, south, west.
Three of the cardinal positions (north, east, and west) lie where the horizon arc is along or perpendicular to the meridian (0° azimuth). While south lies outside the range of the base plate, its general direction will be indicated along the meridian opposite to the north:
Often, these numbers are written small and in Roman numerals to avoid being confused for different values on the base plate.
Southern Hemisphere:
In the Southern Hemisphere, the construction of the points are still based on the position of the horizon arc. However, for the Southern Hemisphere, projection lines are drawn from the north pole instead of the south. The horizon will be mirrored along the base plate, but the general construction will remain the same. And, without labels, the construction would be indistinguishable outside of the flipped position of the horizon arc. 
It’s important to note that the flipped position reverses the numbering of each segment because of where the north and south points are projected to. Unlike for positions in the northern hemisphere, the projection from the north pole pushes the north point out of the base plate. Instead, the south position sits at the midpoint of the horizon arc.
And, the positions for east and west are similarly found with respect to the position of north and south in clockwise order.
However, the numbering is still counted from east to west. So, the final numbers will be counted in a counterclockwise order.